University of Iowa
Iowa Research Online
Theses and Dissertations
1916
Sound waves of finite amplitude
Victor August Hoersch
State University of Iowa
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Recommended Citation
Hoersch, Victor August. "Sound waves of finite amplitude." MS (Master of Science) thesis, State University of Iowa, 1916.
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SOUND WAVES OF FIN ITE AMPLITUDE
by
%
V ic t o r A. Hoersch
A th e s is
s u b m itte d to th e f a c u lt y o f
th e G raduate C o lle g e o f
The
S ta te
U n iv e r s it y
of
Iowa
in p a r t i a l f u l f i l l m e n t o f th e re q u ire m e n ts
f o r th e degree o f
M a s te r
o f
S c ie n c e
August 25 ,19 16 .
SOUND WAVES OF FINITE AMPLITUDE
OUTLINE.
A.
H ydrodynam ica l E q u a tio n s .
I.
II.
B.
System o f E u le r.
System o f Lagrange.
D is c o n t in u it ie s .
I . G eneral
P ro p e rtie s o f D is c o n t in u it ie s and T erm in ology
to be A dopted.
II.
G e n e ra lity o f a D is c o n tin u it y o f th e P ressu re
as a
F u n c tio n W h o lly o f the D e n s ity .
III.
K in e m a tic a l R e la tio n s P e r ta in in g to a D is c o n t in u it y .
IV .
S ta tio n a r y D is c o n t in u it ie s .
V.
D is c o n tin u o u s Waves.
1. D e r iv a tiv e s o f th e D e n s ity .
2. Second and H ig h e r O rder D is c o n t in u it ie s .
3. F i r s t O rder D is c o n t in u it ie s .
a. C o n tin u ity R e la tio n .
b.
Dynam ical E q u a tio n and V e lo c ity F orm ula.
c.
E q u a tio n o f E nergy.
d.
H u g o n io t's Law.
e.
f.
E n tro p y and Thermodynamic P o t e n t ia l.
A p p lic a t io n o f th e E n tro p y C o n d itio n to a
F i r s t O rder D is c o n t in u it y .
g.
h.
4.
M o tio n F o llo w in g an I n i t i a l Plane Wave D is c o n tin u it y .
(1 ).
R iem ann's T re a tm e n t.
(2 ).
Hadamard's T re a tm e n t.
O b je c tio n to H u g o n io t's Law.
Summary.
OUTLINE. - 2 G.
I n f in it e s im a l A m p litu d e Sound Waves.
D.
C ontinuous F i n i t e A m p litu d e Sound Waves.
I.
Plane Waves.
1. G eneral Theory o f C h a r a c te r is tie s f o r Plane Waves.
2. S in g le P ro g re s s iv e Waves.
a. E arnshaw 's C o n d itio n .
b . G eneral S o lu tio n and P ro p e rtie s o f a S u rfa ce I n t e g r a l,
c.
S o lu tio n W ith G iven M o tio n o f P is to n .
d.
S in g u la r it y in th e S o lu tio n .
3. Waves, n o t S in g le P ro g re s s iv e .
a. G eneral S o lu tio n .
b.
A p p lic a tio n to S p e c ia l Cases.
( 1 ) . Riemann’ s S p e c ia l Case.
( 2 ) . L in e a r
Type Waves in r and s.
( 3 ) . L im ite d I n i t i a l D is tu rb a n c e .
II.
S p h e ric a l Waves.
1. I n f in it e s im a l D is tu rb a n c e on a Steady F low .
2. F in it e A m p litu d e Waves.
E. Waves o f Permanent Regime W ith D is s ip a tiv e F o rce s.
I . E n tro p y C o n d itio n .
II.
Heat C o n d u ctio n .
III.
V is c o s it y .
IV .
V is c o s ity and Heat C o n d u ctio n .
F. Summary.
SOUND WAVES OF FINITE AMPLITUDE
The problem o f sound in a gas, in th e most g e n e ra l case,
is a problem o f the most g e n e ra l m o tio n t h a t a gas may have.
We
are le d to a c o n s id e ra tio n o f th e h yd ro d yn a m ica l e q u a tio n s .
These
are the d i f f e r e n t e q u a tio n s w hich a sound wave must s a t is f y .
A.
H ydrodynam ical E q u a tio n s ,1
The h y d ro d yn a m ica l e q u a tio n s a re most g e n e ra lly ex­
pressed in two systems o f r e p r e s e n ta tio n , t h a t o f E u le r
and
th a t o f Lagrange.
I.
System o f E u le r.
In t h is system o f re p re s e n tin g th e m o tio n o f a gas , i f
we adopt r e c ta n g u la r c o o rd in a te s , we lo o k upon th e gas from a system
o f f ix e d re c ta n g u la r axes and examine what happens a t a p o in t whose
c o o rd in a te s are x , y , z.
In d e s c r ib in g what i s happening a t a p o in t
we are in te r e s te d in knowing th e v e lo c it y o f flo w o f th e gas, in
-the eUrisity »f>cl+K€ ^YtiSUrt
m agnitude and d ir e c t io n , a t t h is p o in t .
These q u a n t it ie s w i l l , in
g e n e ra l, v a ry w ith the tim e .
A knowledge o f these q u a n t it ie s a ls o
g iv e s us a com plete knowledge o f th e s ta te o f the gas, f o r th e tem­
p e ra tu re is d e term in ed by th e c h a r a c t e r is t ic e q u a tio n o f the gas
as a fu n c tio n o f th e d e n s ity and p re s s u re ,
homogeneous.
th e gas b e in g assumed
Our d e s c r ip tio n o f th e m o tio n o f the gas, expressed
in m ath e m a tica l language would b e :-
1. I have s e le c te d th e m a te r ia l in t h is s e c tio n from Lamb's H ydro­
dynam ics, 1906 ;C h a p .I, Chap. I I , A r t s . 21 -25 , Chap. X, A r ts . 272-276,
281-282. The dynam ical e q u a tio n f o r a s y m m e tric a l s p h e r ic a l wave is
n o t e x p l i c i t l y g iv e n b u t may be e a s ily o b ta in e d by tra n s fo rm a tio n
o f the g e n e ra l e q u a tio n to s p h e r ic a l c o o rd in a te s .
-2 -
u = f^ ( x ,y ,z , t)
v = f 2(x ,y ,z ,t)
w = f g i 'x , y , z , t )
J° « P ( x , y , z , t )
P=
T T (x ,y ,z ,tj
where u ,v ,w , are the components o f th e v e lo c it y o f flo w a t the
p o in t x , y , z a t th e tim e t in th e d ir e c t io n s o f the x ,y ,a n d z axes
r e s p e c tiv e ly , where
is th e d e n s ity , and where p is th e p re s s u re
a t th e p la c e and tim e .
be p e r f e c t ly a r b i t r a r y .
We can e a s ily see t h a t these fu n d tio n s cannot
f o r exam ple, l e t us im agine any c lo s e d
s u rfa c e f ix e d w ith re s p e c t to the axes o f x ,y ,a n d z.
The flo w o f
gas outward ove r t h is c lo s e d s u rfa c e , d e te rm in e d by th e f i r s t th re e
fu n c tio n s , e v id e n tly has an e f f e c t on the d e n s ity w it h in t h is
s u rfa c e .
T h is in te rde pe nd en ce o f the d e n s ity upon th e v e lo c it y
components is expressed by the s o - c a lle d e q u a tio n o f c o n t in u it y :
(1)
A g a in , we e a s ily see t n a t the p re s s u re cannot be p e r f e c t ly a r b it r a r y
because the gas i s alw ays a c c e le ra te d so as to move from re g io n s
o f h ig h p re s s u re to re g io n s o f lo w e r p re s s u re .
t h is in v o lv e s a re c a lle d
The r e la t io n s w hich
the d yn a m ica l e q u a tio n s .
These a re :
(2)
,
An im p o rta n t case o f the m o tio n o f a gas is r e c t i l i n e a r
t
o r u n ^ ir e c t io n a l m o tio n .
I n t h is case n o t o n ly is the v e lo c it y
o f flo w in one d ir e c t io n b u t i t
is th e same in m agnitude a t e ve ry
p o in t in a p la n e a t r i g h t a n g le s to t h is d ir e c t io n and th e d e n s ity
and p re s s u re are a ls o th e same a t e ve ry p o in t in t h is p la n e .
-3 -
L e t us ta k e the d ir e c t io n o f flo w as t h a t o f the x a x is ,
i'hen e ve ry
q u a n tity in v o lv e d is a f u n c tio n s o le ly o f x and t and v and w
disap-pear from the above e q u a tio n s .
The e q u a tio n o f c o n t in u t iy
thus becomes:
(3)
and we have l e f t th e one d yn am ical e q u a tio n :
(4)
If
, in a d d it io n , a t e ve ry p o in t ,
the c o n d itio n o f th e gas does n o t
v a ry w ith tim e , we have ste a d y r e c t i l i n e a r m o tio n .
u , p , and p a re fu n c tio n s s o le ly o f x .
T h is means th a t
The p a r t i a l d i f f e r e n t i a l
e q u a tio n s the n become o rd in a r y and we o b ta in by in t e g r a t io n :
(5)
(6)
The l a s t e q u a tio n expresses th e c o n s e rv a tio n o f e n e rg y.
The f i r s t
term is the k in e t ic energy o f u n it mass o f the gas and the l a t t e r
its
p o t e n t ia l e n e rg y.
T h is is c a lle d . B e r n o u lli's e q u a tio n .
Besides r e c t i l i n e a r m o tio n , a n o th e r im p o rta n t case o f the
m otio n o f a gas is s y m m e tric a l s p h e r ic a l m o tio n in w h ich th e ve­
lo c it y
is everywhere in th e d ir e c t io n o f the r a d ic a l lin e s from a
f ix e d p o in t w hich i s th e c e n te r o f a system o f spheres upon each
o f w hich th e v e l o c i t y , d e n s ity , and p re s s u re are c o n s ta n t,
i f we
adopt s p h e r ic a l c o o rd in a te s and take the c e n te r p o in t as the p o le ,
a l l the q u a n t it ie s in v o lv e d become fu n c tio n s s o le ly o f r and t .
Then, i f u denote th e v a lu e o f th e r a d ic a l v e l o c i t y ,
th e e q u a tio n
o f c o n t in u it y is :
(7)
and the dynam ical e q u a tio n is :
(8 )
-4 -
I f the m otio n is ste a d y u,^0 , and p a re fu n c tio n s s o le ly o f r ,
th e
above p a r t i a l d i f f e r e n t i a l e q u a tio n s become o r d in a r y and we o b ta in
by in t e g r a t io n :
(9)
(10)
I n case the gas is a cte d upon by a system o f fo rc e
th ro u g h o u t i t s
volum e, we must s u b s t it u t e i n th e dyn am ical
e q u a tio n s f o r -
!
d t ° , iv,
/
5 / where 7) Jo is th e deJ z ~ 'W
r iv A t iv e o f th e p re s s u re in a d ir e c t io n n and Jn th e component o f
the r e s u lt a n t o f a l l th e fo rc e s p e r u n i t mass a c t in g a t a p o in t
x , y , z in th e same d ir e c t io n ,
i ’o r the e q u a tio n s in r e c ta n g u la r
c o o rd in a te s we must s u b s t it u t e in e q u a tio n s ( 2 ) :
(ID
where X,Y ,Z a r e 'th e c o o rd in a te s o f th e r e s u lt a n t o f a l l th e fo rc e s
p e r u n it mass a c tin g a t the p o in t x , y , z a t a tim e t and where each
is supposed to be g iv e n as a f u n c tio n o f x , y , z , t .
To summarize, i have in t h i s s e c tio n g iv e n the d i f f e r e n t i a l
e q u a tio n s o f m o tio n , w h ich must be s a t is f ie d by th e v e l o c i t y , den­
s i t y , and p re s s u re .
These th re e q u a n t it ie s are th e d e te rm in in g
fa c to r s o f the m o tio n o f a gas in the system o f if u le r .
1 have g iv e n
the reduced e q u a tio n s f o r the s p e c ia l cases o f r e c t i l i n e a r and sym­
m e tr ic a l s p h e ric a l m o tio n and have g iv e n the e q u a tio n s o f s te a d y
m o tio n in these two cases,
l have a ls o in d ic a te d th e changes in
the g e n e ra l e q u a tio n s when fo rc e s are a llo w e d to a c t upon th e gas
th ro u g h o u t i t s
volum e.
-5 I I System o f Lagrange
L e t us c o n s id e r an elem ent o f the gas w h ic h , in the i n i t i a l
c o n d itio n was a t the p o in t a , b , c w it h re s p e c t to a f ix e d system o f
r e c ta n g u la r axe s.
A t a tim e t t h i s elem ent o f gas w i l l have moved
to a new p o s it io n x , y , z w it h re s p e c t to th e same system o f axes.
Then i f we d e te rm in e x , y , and z each as a f u n c tio n o f a , b , c , t , we
know the p o s it io n o f e ve ry elem ent o f th e gas a t e v e ry in s t a n t .
If
we a ls o know th e d e n s ity and p re s s u re a t e ve ry p o in t o u r d e s c r ip tio n
o f th e m o tio n is com plete f o r the te m p e ra tu re i s d e te rm in e d by th e
c h a r a c t e r is t ic e q u a tio n o f th e gas as a f u n c tio n o f the d e n s ity and
p re s s u re . We may, th e n , express our d e s c r ip tio n o f the m o tio n
m a th e m a tic a lly in the form :
x = f 1 ( a ,b ,c ,t )
y = f 2(a ,b ,c ,t)
z = f 3 (a ,b ,c , t)
= F ( a ,b ,c ,t)
ρ
p = π (a ,b ,c ,t)
We can e a s ily see t h a t these fu n c tio n s cannot be p e r f e c t ly a r b i t r a r y .
It
is e v id e n t t h a t th e d e n s ity in th e in s ta n ta n e o u s s t a t e , compared
w ith t h a t in the o r i g i n a l s t a t e , v a r ie s w ith the clo se n e ss o f the
p a ckin g w hich the elem ents o f gas undergo and the clo se n e ss o f the
pa ckin g depends on th e fu n c tio n s f]_, f ^ and f ^ .
T h is in te rde pe nd en ce
is expressed by th e e q u a tio n o f c o n t in u it y :
( 12)
-6 Where JD is the d e n s ity in th e o r i g i n a l s ta te a t th e p o in t a , b , c .
A g a in , th e gas is always a c c e le ra te d in th e d ir e c t io n o f tne
p re s s u re g r a d ie n t and hence th e re i s an in te rd e p e n d e n ce between the
p re s s u re d i s t r i b u t i o n and th e m o tio n o f th e gas as expressed by
th e fu n c tio n s f i , f 2 , and f
by the dyn am ical e q u a tio n .
•
T h is in te rd e p e n d e n ce is expressed
I s h a ll here g iv e th d s e q u a tio n i n a
form in v o lv in g b o th th e E u le r ia n and la g ra n g ia n form s o f re p re s a n ta t io n . L e t us use the sym bol^ to denote d i f f e r e n t i a t i o n i n th e f o r ­
mer system and 'b f o r d i f f e r e n t i a t i o n i n th e l a t t e r system .
Then th e
dynam ical e q u a tio n s a re :
da)
D e r iv a tiv e s in th e two systems are r e la t e d by the f o llo w in g fo rm u la e :
(14)
l e t us c o n s id e r th e s p e c ia l case o f r e c t i l i n e a r m o tio n
and take as th e d i r e c t io n o f m o tio n th a t o f th e x a x is .
S ince th e
c o n d itio n o f the gas is th e same a t a l l p o in ts h a v in g th e same abs<iis$a
x a t the same tim e , x is a fu n c tio n s o le ly o f a and t , and y and z
rem ain eq ua l to t h e i r o r i g i n a l v a lu e s b and c r e s p e c t iv e ly . Then th e
e q u a tio n o f c o n t in u it y becomes:
(15)
We have le x t one d yn a m ica l e q u a tio n :
Since a r e l a t io n between d e r iv a tiv e s in the two systems is :
-7 -
We may s u b s t it u t e
in the above e q u a tio n and we o b ta in :
(16)
I n case the gas is a cte d upon by a system o f fo rc e s
th ro u g h o u t i t s
volum e, we must s u b s t it u t e in th e d yn am ical e q u a tio n s :
(17)
where X ,Y ,Z , are th e components o f th e r e s u lt a n t o f a l l th e fo rc e s
p e r u n i t mass a c tin g on an elem ent o f gas whose c o o rd in a te s are
a ,b ,c a t a tim e t and where each is supposed to be g iv e n as a fu n c tio n
o f a ,b ,c ,t.
To summarize, 1 have i n t h is s e c tio n g iv e n th e d i f f e r e n t i a l
e q u a tio n s o f m o tio n , w hich must be s a t is f ie d by the p o s it io n s o f the
p a r t ic le s , the d e n s ity , and th e p re s s u re .
These th re e q u a n t it ie s
are th e d e te rm in in g f a c t o r s o f th e m o tio n o f a gas in th e system o f
Lagrange.
m o tio n .
1 have g iv e n th e reduced e q u a tio n s f o r th e case o f r e e t lin e a r
I have a ls o in d ic a te d th e changes in the g e n e ra l e q u a tio n s
when fo rc e s are a llo w e d to a c t upon the gas th ro u g h o u t i t s
volum e.
-8 B.
D is c o n t in u it ie s
One o r more of the q u a n t it ie s used i n d e s c r ib in g th e m o tio n of
a gas may a t a c e r t a in tim e he d is c o n tin u o u s a t c e r t a in p o in ts o r the
d e r iv a tiv e s o f these q u a n t it ie s may he d is c o n tin u o u s a t a c e r t a in
tim e and a t c e r t a in p o in t s . The d is c o n tin u itie s w h ich I w ish to d is ­
cuss are those t h a t o ccu r a t is o la te d s u rfa c e s w ith o u t s i n g u l a r i t i e s .
I t may appear a t f i r s t s ig h t as i f such d is c o n t in u it ie s a re e x c e p tio n a l
bu t I s h a ll l a t e r show t h a t the occurence o f such d is c o n t in u it ie s i s
g e n e ra l and is o f im p o rta n ce in th e s tu d y o f th e m o tio n o f a gas. A
sep ara te a n a ly s is must he made as the e q u a tio n s o f m o tio n do n o t a p p ly
to th e d is c o n t in u it y .
I.
G eneral P ro p e rtie s o f D is c o n t in u it ie s and Te rm in o lig y
to he Adopted °
L e t ϕ be some q u a n t it y in v o lv in g th e c o n d itio n o f th e gas
w hich i s d is c o n tin u o u s a t a s u rfa c e , whose e q u a tio n in i n i t i a l c o o rd in a te s
is :
f(a ,b ,c ,t)
= 0
(18)
and in term s o f in s ta n ta n e o u s c o o rd in a te s is :
f 1( x , y , z , t )
= 0
(19)
L e t us te rm th e two re g io n s on e it h e r s id e o f t h is s u rfa c e th e re g io n s
1 and 2 .
Now ϕ
is c o n tin u o u s in b o th th e se re g io n s and th e r e fo r e as
we approach a p o in t on the s u rfa c e o f d is c o n t in u it y by any p a th w hich
l i e s w h o lly in re g io n 1, ϕ
c a ll ϕ 1 .
approaches a d e f in i t e l i m i t w hich we s h a ll
L ik e w is e as we approach a p o in t on the s u rfa c e o f d is c o n t in u it y
° Lecons s u r la H ro p o g a tio n des Ondes a t le s E o u a tio n s de I 'H y d r o dynamiaue, 1903; pp . 83 -85 , 87.
-9 by any p a th w h ich l i e s w h o lly in re g io n 2, ϕ
w h ich we s h a ll c a l l
ϕ 2 . Thusϕ
v a rie s c o n tin u o u s ly u n t i l we rea ch
th e s u rfa c e o f d is c o n t in u it y where i t
tude
approaches a d e f in i t e l i m i t
ta ke s a sudden jump o f m agni­
- ϕ 2 , w hich we s h a ll d e s ig n a te by th e symbol [ ϕ ] a f t e r w n ich
1
ϕ
i t a g a in v a r ie s c o n tin u o u s ly .
Now two p o in t s , P and Q, on the s u rfa c e o f d is c o n t in u it y
may be connected by pa th s i n re g io n s 1 and 2 w h ich d i f f e r by a d is ta n c e
as s m a ll as we p le a se from a p a th ly i n g in th e s u rfa c e o f d is c o n tin ­
u ity .
L e t th e two va lu e s o f ϕ a t P be ϕ 1 P , ϕ 2 P and the two va lu e s
a t Q be ϕ1Q, ϕ2Q. A lo ng th e p a th in re g io n 1 ϕ changes c o n tin u o u s ly
fro m ϕ1P to ϕ1Q and a lo n g th e p a th i n r e g io n 2 <js changes c o n tin u o u s ly
from ϕ2P t o ϕ 2 Q .
We may choose o u r p a th s such t h a t as the p o in t Q
i s made to approach the p o in t P a lo n g a p a th in t he s u rfa c e o f
d is c o n t in u it y , the le n g th s o f each o f these p a th s approaches z e ro .
As the le n g th s o f the se p a th s approach z e ro , s in c e f )
upon each o f them, <jf ^ must approach j)t ^ and
Thus
<f)t
is c o n tin u o u s
p must approach j %^
•
, must be a c o n tin u o u s f u n c tio n on th e s u rfa c e o f d is c o n ­
t i n u i t y and a ls o
must be c o n tin u o u s on t h is s u rfa c e .
The L a g ra n g ia n system o f r e p re s e n ta tio n is most u s e fu l
in the s tu d y o f d i s c o n t i n u it ie s .
The q u a n t it ie s x , y , z o r any o f t h e i r
d e r iv a tiv e s in th e la g ra n g ia n system o f c o o rd in a te s may be d is ­
co n tin u o u s when th e se v a lu e s are a ssig n e d to t h e i r p o s itio n s i n a
sphere whose c o o rd in a te s are a , b , c .
As a m a tte r o f te rm in o lo g y ,
we s h a ll term th e o rd e r o f th e d is c o n t in u it y th e o rd e r o f th e lo w ­
e s t d e r iv a tiv e w hich s u f f e r s a d is c o n t in u it y .
is o f the n ‘ th o rd e r and i f
t h is
Thus
3 ^ X_______
is th e lo w e s t o rd e r d e r iv a t iv e w hich
is d is c o n tin u o u s a t the d is c o n t in u it y , th e d is c o n t in u it y is o f th e n 't h
o rd e r.
(1 0 )
To sum m arize, we have pro ven an im p o rta n t p r o p e r ty o f a
d is c o n t in u it y ,
the c o n t in u it y o f the two v a lu e s th e re assumed ove r
the s u rfa c e o f d is c o n t in u it y .
T h is a llo w s us to form a t o t a l de­
r iv a t i v e o f e it h e r o f these v a lu e s on th e s u rfa c e .
We have a ls o i n ­
d ic a te d the arrangem ent o f d is c o n t in u it ie s a c c o rd in g to t h e i r o rd e r.
I I . G e n e r a lity o f th e E x is te n c e o f a D is c o n t in u it y i f
th e
P ressure is a F u n c tio n w h o lly o f th e D e n s ity 0
I f in a gas, th ro u g h o u t i t s
supposed to d i s t r i b u t e
m o tio n , h e a t developed is
i t s e l f im m e d ia te ly so t h a t th e gas is always
a t a c o n s ta n t te m p e ra tu re , th e p re s s u re o f t h e gas a t e ve ry p o in t
is p r o p o r tio n a l to the d e n s ity , in accordance w ith Bo y le 's law :
(20)
If,
on th e o th e r hand, th e h e a t c o n d u c t iv it y o f th e gas may he con­
s id e re d z e ro , the change o f s t a te o f each elem ent o f gas is a d ia b a tic
and th e p re s s u re v a rie s as th e d e n s ity to t h e Y power where Y = Sp/S
v
is the r a t i o
o f th e s p e c if ic h e a t a t c o n s ta n t p re s s u re to t h a t a t
c o n s ta n t volume, i n accordance w ith P o is s o n 's a d ia b a tic la w , nam ely:
(21)
We may w r it e to ta ke care o f e it h e r case:
C22)
Now in any m ech an ica l system , th e d i f f e r e n t i a l e q u a tio n s of m o tio n
a llo w us to c a lc u la te th e a c c e le r a tio n s o f i t s
component p a r t ic le s
a t any in s t a n t when we have g iv e n the p o s it io n s o f a l l th e se p a r­
t i c l e s Land t h e i r v e lo c it ie s a t t h a t in s t a n t s u b je c t to th e c o n d itio n
° Hadgmard, lo c , c i t .
pp . 139, 142.
(11)
th a t the p o s itio n s o f th e d i f f e r e n t p a r t ic le s and t h e i r v e lo c it ie s
must s a t is f y the c o n s tr a in ts imposed upon th e system .
m echanical system w ith an i n f i n i t e
number o f degrees o f freedom . In
the case o f th e m o tio n o f a gas, e q u a tio n s (1 2 ),
g iv e ilipon e lim in a t io n
o f p and
A gas is a
(13) and (E2)
the components o f a c c e le r a tio n
, (Y*-^ i f we are sciven x ,y ,a n d z as fu n c tio n s o f a ,b ,c
'S t*
a t th e tim e a t w h ich we are making ou r o b s e rv a tio n and a ls o
»
3 i:
d V _, ?) ^ , as fu n c tio n s o f a ,b ,e a t th e same tim e .
The c o n s t r a in t imposed upon th e gas is t h a t a t th e bounding
w a ll, th e elem ents o f gas in c o n ta c t w ith th e w a ll must move so as
to rem ain always i n c o n ta c t w ith t h is w a ll. We s h a ll he re exempt from
d is c tis s io n a t t h is tim e eases where th e v e lo c it y o f th e bounding w a ll
is oneof decom pression and o f such r a p i d i t y th a t th e gas is n o t a b le
to f o llo w the bounding w a ll b u t le a v e s a vacuum between th e s u rfa c e
o f th e bounding w a ll and th e s u rfa c e o f th e gas.
L e t th e e q u a tio n
o f th e bounding s u rfa c e be:
f
(x ,y ,z ,t)
* 0
(23)
S ince a p a r t i c l e , whose i n i t i a l c o o rd in a te s are a ,b ,c and w h ich o r i g i n a l ly
l i e s on t h is s u rfa c e , must c o n tin u e to l i e
on t h is s u rfa c e , by d i f ­
f e r e n t ia t in g th e e q u a tio n o f t h is s u rfa c e w it h re s p e c t to tim e , we o b ta in
as a ne cessa ry c o n d itio n upon th e g iv e n v a lu e s o f th e v e lo c it y a t
p o in ts on the boundary
the e q u a tio n :
(24)
We suppose th a t the g iv e n va lu e s o f th e v e lo c it y a t the boundary s a t is f y
t h is e q u a tio n .
D i f f e r e n t i a t i n g once more w it h re s p e c t to tim e we o b ta in :
(25)
-1 2 Now th e components o f a c c e le r a tio n are a lre a d y d e term in ed by the
e q u a tio n s o f m o tio n (13) th ro u g h o u t th e volume and hence upon th e bound­
in g s u rfa c e .
The compnnents o f a c c e le r a tio n th u s de term in ed w i l l n o t,
in g e n e ra l, s a t i s f y t h is e q u a tio n i f th e m o tio n o f th e bounding w a ll,
and hence th e f u n c tio n f , may be c o n s id e re d as a r b i t r a r i l y chosen.
The c o n t r a d ic t io n here in v o lv e d may be re s o lv e d by th e f o llo w in g con­
c e p tio n .
We may approach the boundary a lo n g a p a th ly i n g w h o lly in the
enclosed gas.
As we approach th e boundary a lo n g t h i s p a th by a d is ta n c e
as s m a ll as we p le a se , th e a c c e le r a tio n is g iv e n th ro u g h o u t by th e
e q u a tio n s o f m o tio n (13) b u t th e p a r t i c l e a t th e te r m in a l p o in t on th e
boundary has an a c c e le r a tio n w h ich s a t is f ie s e q u a tio n (25)
S ince
a re p r o p o r tio n a l to th e d ir e c t io n c o sin e s o f th e norm al
S t b
2
to the* s u rfa c e (19)
we see t h a t e q u a tio n (25) a f f e c t s o n ly th e norm al
component o f the a c c e le r a tio n .
if
the norm al component o f a c c e le r a tio n
o b ta in e d as a l i m i t by approach a lo n g a p a th fro m w it h in the gas does
n o t agree w ith the va lu e o b ta in e d fnrom e q u a tio n (2 5 ), th e re e x is ts a
d is c o n t in u it y a t th e bounding s u rfa c e ,
, b e in g the no rm al component
o f a c c e le r a tio n , o b ta in e d from e q u a tio n (25) and (p^ b e in g th e v a lu e o f
t h is component o b ta in e d as a l i m i t by approach fro m w it h in th e gas.
Thus in t h is case a second o rd e r d is c o n t in u it y is generated and t h is
w i l l be propogated in t o th e gas a c c o rd in g to c e r t a in laws w h ich we
s h a ll f u r t h e r in v e s t ig a t e .
L e t us suppose t h a t the va lu e s o f th e a c c e le r a tio n o b ta in e d
from e q u a tio n s (13) agree w ith e q u a tio n (25)
have f u r t h e r c o n d itio n s to be f u l f i l l e d .
on the boundary. We then
When th e p o s it io n s and ve -
1
D i f f e r e n t i a l and in t e g r a l C a lc u lu s - G r a n v ille , 1911: p . 266
-1 3 -
lo c it ie s
o f the p a r t ic le s are g iv e n n o t o n ly th e second d e r iv a tiv e s o f
x , y , and z may he o b ta in e d b u t e ve ry d e r iv a t iv e .
The second d e r iv a t iv e s ,
∂2x/∂a2, ∂2x/∂a∂b, ∂2x/∂a∂c, ∂2x/∂b2, -----, ∂2z/∂c2 a re known because x , y , and z
are g iv e n as fu n c tio n s o f a , b , c .
The second d e r iv a t iv e s , ∂2x/∂a∂t, ∂2x/∂d∂t,
----------- , ∂2z/∂b∂t, ∂2z/∂c∂t a re known because ∂ x / ∂ t , ∂ y / ∂ t , and ∂ z / ∂ t are
g iv e n as fu n c tio n s o f a , b , c .
As shown in th e p re c e e d in g pa ra g ra p h th e
second d e r iv a t iv e s , ∂2x/∂t2, ∂2y/∂t2 and ∂ 2 z /∂ t2 a re known fro m the e q u a tio n s
o f m o tio n (1 3 ).
o f a ,b ,c .
Thus a l l th e second d e r iv a tiv e s are known as fu n c tio n s
T his a llo w s us to c a lc u la te
by d i f f e r e n t i a t i o n a l l the
t h i r d d e r iv a tiv e s e xce p t ∂ 3 x /∂ t3 , ∂ 3 y /∂ t3 and ∂ 3 z /∂ t3 .
By d i f f e r e n t i a t i n g
each o f e q u a tio n s (13) w ith re s p e c t to t we o b ta in these d e r iv a tiv e s
and hence a l l the t a i r d d e r iv a tiv e s are known.
E q u a tio n (25) may be
d if f e r e n t ia t e d w ith re s p e c t to t f o r a c o n d itio n on the boundary and
i f t h is r e l a t io n is n o t s a t is f ie d by the t h i r d o rd e r tim e d e r iv a tiv e s
determ ined fro m th e g iv e n s ta te o f th e gas, th e re is a t h ir d
d is c o n t in u it y a t th e bounding s u rfa c e .
o rd e r
A l l th e f o u r t h o rd e r d e r iv a tiv e s
except ∂4x/∂t4, ∂4y/∂t4, and ∂4z/∂t4 a re found by d i f f e r e n t i a t i n g th e t h i r d
o rd e r
d e r iv a tiv e s w it h re s p e c t to a ,b ,c and hence are known, w h ile th e se de­
r iv a t iv e s are known by d i f i e r e n t i f c t i n g tw ic e e q u a tio n s (1 3 ).
T his
process can o b v io u s ly be re p e a te d and hence a l l th e d e r iv a tiv e s o f x ,
y , and z are known.
L ik e w is e by d i f f e r e n t i a t i n g s u c c e s s iv e ly e q u a tio n
(25) we o b ta in added c o n d itio n s upon the se d e r iv a tiv e s so t h a t even
sho uld th e t h i r d d e r iv a tiv e s a t the boundary as d e te rm in e d from th e g iv e n
s ta te agree w ith the boundary c o n s t r a in t , th e f o u r t h o rd e r d e r iv a tiv e s
m ig h t n o t agree and we sh o u ld have a f o u r t h o rd e r d is c o n t in u it y .
Thus,
in g e n e ra l, an a r b it r a r y m o tio n o f th e boundary im p lie s a d is c o n t in u it y
in the gas.
In th e f u r t h e r p ro g re s s o f th e m o tio n beyond th e g iv e n s ta te
should th e boundary w a lls d e s c rib e a m o tio n d is c o n tin u o u s a t a c e r t a in
-1 4 time t in th e n 't h o rd e r d e r iv a t iv e s , an n ’ th o rd e r d is c o n t in u it y w i l l be
formed in th e gas.
To summarize, i f
th e p o s itio n s and v e lo c it ie s o f th e p a r t ic le s
o f gas are g iv e n so as to be c o m p a tib le w ith the p o s it io n and v e lo c it y
o f the bounding w a lls , th e a c c e le r a tio n and tim e d e r iv a tiv e s o f h ig h e r
o rd e r w i l l n o t, in g e n e ra l, be c o m p a tib le w it h the m o tio n o f the
bounding w a lls and d is c o n t in u it ie s o f the second o r h ig h e r o rd e r
w i l l ensue.
III
K in e m a tic a l R e la tio n s P e r t a in in g to a D is c o n t in u it y
I w is h f i r s t to ta ke up th e v e lo c it y o f p ro p a g a tio n
o f a d is c o n t in u it y in two systems o f measurement.
The one system o f
measurement has to do w ith th e e q u a tio n o f th e s u rfa c e o f
d is c o n t in u it y in th e fo rm :
f
(a ,b ,c ,t ) = 0
(26)
and the o th e r in th e fo rm :
ϕ (x ,y ,z ,t)
=
0
L e t us f i r s t c o n s id e r th e fo rm e r system .
(27 )
In t h i s system
we im agine th e v a lu e s x , y , z a ssig n e d to e v e ry p o in t in a space whose
c o o rd in a te s are a , b , c .
The d is c o n t in u it y i s s it u a t e d on a moving
s u rfa c e in t h i s space whose e q u a tio n we may re p re s en t by (2 6 ).
We
may im agine a p o in t a ,b ,c w h ich moves w it h a v e l o c i t y alw ays norm al
to th e s u rfa c e ( 3 f ) and so as to rem ain alw ays in t h i s s u rfa c e .
The
■V elocity o f t h i s p o in t we s h a ll c a l l th e v e lo c it y o f pro pa g a tio n o f
th e d is c o n t in u it y w it h re s p e c t to th e i n i t i a l s ta te a t th e p o in t
c o n s id e re d .
We s h a ll d e s g in a te t h i s v e lo c it y by th e l e t t e r θ
1. Hadama^d, lo c . c i t . , pp. 101-105.
I have e la b o ra te d on th e p r o o f o f
the fo rm u la f o r th e v e lo c it y o f p ro p o g a tio n w h ich Hadamard o n lv
b r i e f l y o u t lin e s .
-1 5 -
The li n e
jo in in g a p o in t a ', b ' , c ' o u ts id e the s u rfa c e o f d is c o n t in ­
ui t y to th e p o in t a, b , c on t h i s s u rfa c e , c o n s id e re d as a v e c to r ,
has components, a - a ', b - b ', c- c ' ,
The p r o je c t io n o f t h i s l i n e segment
upon th e norm al a t a , b , c i s th e sum o f th e p r o je c tio n s of these components
on
th e n o rm a l. To fo rm th e p r o je c t io n
o f any component we must m u lt ip ly
i t by th e d ir e c t io n co sin e o f th e norm al w it h re s p e c t to th e c o o rd in a te
a xis c o rre s p o n d in g to t h i s component. Now th e d ir e c t io n c o s in e s o f th e n o r­
mal a re :
si n ce th e y are p r o p o r tio n a l t o ∂f/∂a, ∂f/∂b, ∂f/∂c1 and s in c e th e sum o f t h e i r
squares must e q u a l one.
Then the p r o je c t io n o f the li n e
segment upon
the norm al i s :
(28)
If th e p o in t a ' , b ’ , c ' is f ix e d and th e p o in t a ,b ,c is moving w ith a ve­
lo c it y whose components are d a /dt, db/dt, dc/dt, th e n the r a t e o f change o f th e
normal component o f th e l i n e
segment i s o b ta in e d by d i f f e r e n t i a t i n g th e
above e x p re s s io n f o r n , nam ely:
1.
D i f f e r e n t i a l and I n t e g r a l C a lc u lu s - G r a n v ille , 1911 p . 266.
-1 6 Now when th e p o in t a ' , b ' c ' c o in c id e s w it h th e p o in t a , b , c , i t
is ob­
v io u s th a t d n /d t i s th e v e l o c i t y o f th e p o in t a , b , c such as we have
d e s c rib e d i t
and hence i s e q u a l t o th e v e l o c i t y o f p ro p o g a tio n θ a t
t h is p o in t .
D i f f e r e n t ia t in g e q u a tio n (25) o f th e s u rfa c e , we o b ta in
f o r th e p o in t a , b , c , s in c e i t
lie s
on t h i s
s u rfa c e , th e ne ce ssa ry con­
d itio n :
(30)
S u b s t it u t in g th e sum o f th e f i r s t th r e e term s o b ta in e d fro m t h i s equ­
a tio n in th e f i r s t te rm o f th e r i g h t hand member o f (29) and p la c in g
a = a ’ , b = b ' , c = c ', and dn/dt = θ we o b ta in :
(31)
We s h a ll adopt th e fo rro w in g c o n v e n tio n as t o s ig n . The f u n c t io n
f i n the l e f t member o f (26) s h a ll be p o s it iv e i n re g io n 2. The
v e lo c it y o f p ro p a g a tio n s h a ll be c o n s id e re d p o s it iv e o r n e g a tiv e
a c c o rd in g as th e s u rfa c e o f d is c o n t in u it y moves fro m re g io n 1 in t o
re g io n 2 o r i n th e in v e rs e d ir e c t io n .
By these same methods we may f i n d th e v e lo c it y o f p ro p a g a tio n
T o f the s u rfa c e o f d is c o n t in u it y w it h re s p e c t to th e a c tu a l s ta te
t
m e re ly u s in g e q u a tio n (27) o f th e s u rfa c e and c o o rd in a te s x , y , z
in s te a d o f a , b , c . The v e l o c i t y o f p ro p o g a tio n T i s th e in s ta n ta n e o u s
v e lo c it y o f a p o in t in th e s u rfa e e
norm al to th e s u rfa c e . F o r t h i s
v e lo c it y we have a v a lu e analogous t o Q i n e q u a tio n ( 3 1 ) , nam ely
(32)
The v e lo c it ie s T
and #
a re d i s t i n c t
even i f th e i n i t i a l s ta te
-1 7 c o in c id e s w it h the a c tu a l s ta te a t th e in s t a n t c o n s id e re d ,
case x , y , z are r e s p e c t iv e ly e q u a l to a ,b ,c a t t h is
ϕ
and f are eq u a l a t t h is
in s t a n t .
how in t h i s
in s t a n t and henc e
T h e re fo re ∂ϕ /∂x, ∂ϕ /∂y, ∂ϕ /∂z are
r e s p e c tiv e ly eq u a l to ∂ f/∂ a , ∂ f/∂ b , ∂ f/∂ c . On th e o th e r h a n d we have:
(33)
where u ,v ,w are components o f th e v e l o c i t y .
D iv id e t h i s e q u a tio n
th ro u g h o u t by √((∂ϕ/∂x)2+(∂ϕ/∂y)2+(∂ϕ/∂z)2) o r i t s e
c.
b
+
(∂f/)2
l√
a
u
q
Then by v ir t u e o f the s t atem ent p ro c e e d in g e q u a tio n (2 8 ), th e f i r s t
th re e term s o f the r i g h t hand member o f e q u a tio n (35) become the norm al
component o f th e v e lo c it y a t th e d is c o n t in u it y w hich we s h a ll d e s ig n a te
by 1^,
.
A lso u s in g e q u a tio n s (31) and (3 2 ), e q u a tio n (33) becomes:
or
(34)
The p r i n c i p a l o f c o m p o s itio n o f v e lo c it ie s would a ls o serve to g iv e us
t h is r e s u lt f o r when the i n i t i a l s ta te is ta ke n c o in c id e n t w ith the
a c tu a l s t a t e , th e v e lo c it y 0 is th e v e lo c it y o f p ro p a g a tio n o f th e s u r­
face o f d is c o n t in u it y w ith re s p e c t to th e medium in i t s
a c tu a l s ta te and
the a c tu a l v e lo c it y o f p ro p a g a tio n T may be c o n s id e re d as th e r e s u l t ­
a n t o f th e v e lo c it y o f p ro p a g a tio n 0 w it h re s p e c t to th e medium and the
v e lo c it y , UVi o f the medium i t s e l f .
Under th is head I w is h a ls o to d is c u s s th e k in e m a tic a l r e l a t io n
e x is t in g between th e s e v e ra l n ' t h o rd e r d e r iv a tiv e s in an n 't h o rd e r
d is c o n t in u it y ,
l e t <J) be a q u a n t it y w hich has an n ’ t h o rd e r d is c o n ­
t i n u i t y a t a s u rfa c e re p re s e n te d by e q u a tio n ( 2 $ ) .
l s h a ll d e s ig n a te
by s u b s c rip ts 1 and 2 th e v a lu e s o f q u a n t it ie s p e r t a in in g to th e r e ­
g io n s 1 and 2 on e it h e r s id e o f th e s u rfa c e o f d is c o n t in u it y .
Then
agree in v a lu e and so do c o rre s p o n d in g d e r iv a tiv e s o f o rd e r
le s s tha n n o f these q u a n t it ie s .
Thus we have.
-1 8 -
on the s u rfa c e o f d is c o n t in u it y f o r p , q , r , h , a n y p o s it iv e in te g e rs
whose sum is n - 1 . D i f f e r e n t ia t in g t o t a l l y w ith re s p e c t to tim e ,
we o b t a in : -
Now a c c o rd in g to o u r assum ption o f th e e x is te n c e o f an n' th o rd e r
d is c o n t in u it y , s im i la r l y p la c e d term s in th e two members o f t h is equa­
t io n are n o t, in g e n e ra l , e q u a l.
B rin g in g a l l the term s to one s id e
o f the e q u a tio n we may w r ite :-
where the b ra c k e ts denote th e v a r ia t io n s o f the q u a n t it ie s th e y enclose
a t the s u rfa c e o f d is c o n t in u it y .
Now e q u a tio n (3 0 ) must lik e w is e h o ld
s in c e the t o t a l d i f f e r e n t i a t i o n can o n ly a p p ly on th e s u rfa c e (26) .
D iv id in g the e q u a tio n (30) th ro u g h o u t by √((∂f/∂a)2 +(∂t/∂b)2 +(∂f/∂c)2) and making use
o f the sta te m e n t p re c e d in g (28) and o f e q u a tio n (3 1 ), we o b t a in : (36)
w h e re α
γare th e d ir e c t io n co sin e s o f th e norm al a t the p o in t c o n s id e re d .
,β
S ince da/dt, db/dt, dc/dt, in e q u a tio n s (35) and (36)
s t i l l rem ain a r b i t r a r y ,
the c o e f f ic ie n t s o f the se q u a n t it ie s and th e n o n - a r b itr a r y term must be
p r o p o r tio n a l in these two e q u a tio n s and th e r e fo r e we h a v e :-
S ince the r e s u lt i s v a lid f o r p , q , r , h any p o s it iv e in te g e r s whose
sum is n - 1, we may w r ite : (37)
P la c in g ϕ s u c c e s s iv e ly e
q
u
a
l to x , y , z and l e t t i n g th e v a lu e s o f the
c o n s ta n t in (37) b e λ , μ , ν , r e s p e c t iv e ly , we o b t a in : -
(38)
As a s p e c ia l case we have : -
-1 9 -
(39)
The l e f t members o f the above e q u a tio n s a re components o f a v e c to r
∂ η s /∂ tη and hence λ,μ
ν
a re components o f a v e c to r c o in c id e n t w ith t h is
v e c to r and c a lle d by Hadamard th e c h a r a c t e r is t ic segment.
Now we
see from (38 ) th a t as f a r as the v a r ia t io n i n th e n 't h d e r iv a tiv e s are
concerned, the d is c o n t in u it y depends o n ly upon th e c h a r a c t e r is t ic
segment and th e v e lo c it y 0
To summarize, I have developed fo rm u la e f o r the v e lo c it y
o f p ro p a g a tio n o f a d is c o n t in u it y w ith re s p e c t to the i n i t i a l and
a c tu a l s ta te o f the gas and have shown th e r e l a t io n between these two
v e lo c it ie s when th e i n i t i a l s ta te c o in c id e s w ith th e a c tu a l s ta te
a t th e in s ta n t c o n s id e re d .
I have a ls o developed n e ce ssa ry r e la t io n s
between th e n ' t h o rd e r d e r iv a tiv e s in an n ' t h o rd e r d is c o n t in u it y
and have shown th a t a d is c o n t in u it y is c h a ra c te riz e d by a c h a ra c te r­
is tic
segment a t e v e ry p o in t on i t s
s u rfa c e and a v e lo c it y o f p ro ­
p o g a tio n 0 a t e ve ry p o in t .
IV
I V . S ta tio n a r y D is c o n t in u it ie s 1
A s ta tio n a r y d is c o n t in u it y is one w h ich a f f e c t s alw ays th e
same p a r t ic le s , t h a t i s , a d is c o n t in u it y f o r w h ich e q u a tio n (26)
does n o t in v o lv e t o r f o r w hich
th e r e fo r e , th a t 0 is
z e ro .
b ± » 0 . We see from e q u a tio n (3 1 ),
Z>t
P la c in g 0 * o in e q u a tio n s (38) we see th a t
[∂ηx/(∂ad∂bk∂cl∂tm)], [∂ηy/(∂ad∂bk∂cl∂tm)], [∂ηz/(∂ad∂bk∂cl∂tm)] a re zero u n le s s m = o .
LdoiJ
^
I
Thus we see th a t ^ f o r a s t a t io n a r y d is c o n t in u it y , th e f i r s t d e r iv a tiv e s
th a t are d is c o n tin u o u s are those n o t in v o lv in g t .
the n ' t h o rd e r d e r iv a tiv e s
1. Hadamard
lo c . c i t . ,
T^x >
,
~ T r» '
MV
1903, p p .99-101
2 )\:
We th u s see th a t
can o n ly be d is c o n -
-2 0 tin u o u s i f x , y , z are d is c o n tin u o u s on th e s u rfa c e (2 6 ),
Then we may
ta ke two p o in ts a1, b1, c1 and a2 ,b 2 c 2 as c lo s e to g e th e r as we p le a s e
and se p a ra te d by the s u rfa c e o f d is c o n t in u it y , w h ile t h e i r a c tu a l
p o s itio n s x i , y _ , z_ and x , y g Zg a re a f i n i t e
d is ta n c e a p a r t,
le t
us c o n s id e r th e e q u a tio n o f the s u rfa c e in a c tu a l c o o r d in a te s , ( 2 7 ) .
Then s in c e the d is c o n t in u it y a fr e e ts alw ays th e same p a r t i c l e s , x^
y^ z^ and x g y g Zg must s a t is f y t h is e q u a tio n th ro u g h o u t th e m o tio n .
Hence a s t a t io n a r y d is c o n t in u it y re p re s e n ts a m o tio n o f th e gas on
e it h e r s id e o f th e d is c o n t in u it y as i f th e s u rfa c e o f d is c o n t in u it y
was a bounding s u rfa c e f o r the two masses o f gas on e it h e r s id e .
We
thu s see th a t th e v e lo c it y must alw ays be t a n g e n tia l to t h i s s u rfa c e .
Y.
D is c o n tin u o u s Waves
A wave ty p e o f d is c o n t in u it y i s one w h ich is p ro p o g a te d from
one p a r t ic le
to a n o th e r, t h a t i s , one f o r w h ich 6 is n o t z e ro , and hence
t h is typ e o f d is c o n t in u it y and th e s t a t io n a r y typ e are m u tu a lly ex­
c lu s iv e .
A d is c o n t in u it y in x , y j
and z i s th e r e fo r e e lim in a te d from
d is c u s s io n and we see fro m e q u a tio n s (38) th a t a d is c o n t in u it y o f the
wave type always a f f e c t s the tim e d e r iv a tiv e s o f x , y , z .
1. D e r iv a tiv e s o f th e n e n s ity 1 .
The d e n s ity is d e fin e d by the e q u a tio n o f c o n t in u it y (12)
I t i s , a c c o r d in g ly , a f u n c tio n o f th e f i r s t d e r iv a tiv e s o f x , y , and z
w ith re s p e c t to a , b , c .
A d is c o n t in u it y in the se f i r s t o rd e r d e r iv a tiv e s
pro du ces, in g e n e ra l, a d is c o n t in u it y in th e d e n s ity .
A t an n 't h o rd e r
d is c o n t in u it y , t h e ( n - l ) ’ 6ii o rd e r d e r iv a tiv e s o f th e d e n s ity a re the
d e r iv a tiv e s f i r s t a f f e c t e d .
E q u a tio n (12) is :
1. Hadamard lo c . c i t . pp. 7 0 .7 1 .1 1 3 .
-2 1 l e t us a tte m p t to fo rm th e v a r ia t io n o f an ( n - l ) s t o rd e r d e r iv a t iv e
of
ρ0/ρ a t a s u rfa c e o f d is c o n t in u it y , t h a t is :
(40)
where th e f i r s t term i n th e r i g h t hand member o f t h is e q u a tio n is an
( n - l ) s t o rd e r d e r iv a t iv e by approach fro m th e re g io n 1 and the second
term th e same d e r iv a t iv e by approach fro m re g io n 2 .
In fo rm in g the se
two term s by d i f f e r e n t i a t i n g e q u a tio n (1 2 ), term s c o n ta in in g o n ly
d e r iv a tiv e s o f x , y o r z o f o rd e r le s s th a n n may be o m itte d s in c e the
value s o f th e se d e r iv a tiv e s by approach from re g io n s 1 and 2 agree and
hence th e y d is a p p e a r fro m e q u a tio n (4 0 ).
The re m a in in g term s are made
up o f the ( n - l ) s t d e r iv a t iv e o f each elem ent m u lt ip lie d by the m in o r
o f th a t elem ent w ith the a lg e b r a ic s ig n c o rre s p o n d in g to th e p o s it io n
o f the elem ent in th e d e te rm in a n t.
A s im p l i f i c a t i o n i s in tro d u c e d
i f we choose as i n i t i a l c o o rd in a te s the a c tu a l c o o rd in a te s in re g io n
2 a t the in s t a n t c o n s id e re d .
Then in re g io n 2, x= a , y= b , z=c and these
e q u a tio n s may be d i f f e r e n t i a t e d w it h re s p e c t to a , b ,c f o r a l l h ig h e r
d e r iv a tiv e s n o t in v o lv in g t .
I n re g io n 2 we th e n have ∂x/a
,
zc1
yb
=
∂x/∂b = ∂x/∂c = ∂y/∂a = ∂y/∂c = ∂z/∂a = ∂z/∂c = o and a l l h ig h e r d e r iv a tiv e s w it h r e ­
sp e ct to a ,b ,c z e ro .
The d e r iv a tiv e s o f o rd e r le s s than n on th e d is c o n t in u it y by
approach from re g io n 1 agree w it h th e c o rre s p o n d in g d e r iv a tiv e s by
approach fro m re g io n 2 . The e v a lu a tio n ju s t made f o r th e d e r iv a tiv e s in
re g io n 2 thus h o ld s f o r d e r iv a tiv e s o f o rd e r le s s th a n n on th e d is ­
c o n t in u it y by approach fro m re g io n 1. l o r a d is c o n t in u it y o f the second
o r h ig h e r o rd e r we th u s have
∂=
y
c
a
z
b
/
o
∂ x /∂ a = ∂ y /∂ b = ∂ z /∂ c = 1 and ∂x/∂b = ∂x/∂c =
and the d e te rm in a n t D becomes one, a l l
elem ents e xce pt those on th e p r in c ip a l d ia g o n a l becoming one t
o f an elem ent n o t on th e p r i n c i p a l d ia g o n a l is zero w h ile th e
|| ~~
The m inor
m in o r o f
A,p"i
-2 2 a p r in c ip a l elem ent is one.
Thus th e o n ly term s in the r i g h t hand
member o f (40) re m a in in g are t h e ( n - l ) s t d e r iv a tiv e s o f th e th re e p r i n ­
c ip a l elem ents m u lt ip lie d by t h e i r re s p e c tiv e m in o rs w h ich are 1.
Thus
e q u a tio n (40) becomes:
S u b s t it u t in g th e va lu e s of the v a r ia t io n s in the r i g h t hand member from
e q u a tio n s (3 8 ), we have:
(41)
Now any fu n c tio n o f ρ may be expressed in the form F ( ρo/ρ) and an
( n - l ) s t d e r iv a t iv e o f t h is f u n c tio n is F'1-(ρo/ρ) m u lt ip lie d by th e co­
re sp o n d in g d e r iv a t iv e
I f we assume an
o f (ρo/ρ), t h a t is
:
i n i t i a l s ta te c o in c id e n t w ith th e a c tu a l s ta te in
re g io n 2 we have f o r a second o r h ig h e r o rd e r d is c o n t in u it y
ρ = ρ0 and we may the n ta k e v a r ia t io n s a t the d is c o n t in u it y o f the
above q u a n t it y . We th u s o b ta in , m aking use o f e q u a tio n (4 1 ):
2.
Second and H ig h e r O rder Dis c o n t in u i t ies1
L e t us le a v e d is c o n t in u it ie s o f th e f i r s t o rd e r f o r l a t e r
d is c u s s io n .
The e q u a tio n s (13) h o ld v a l id f o r th e re g io n s 1 and 2.
S u b s t it u t in g the va lu e o f p from (1 8 ), we o b ta in :
(43)
Now f o r a second o r h ig h e r o rd e r d is c o n t in u it y , i f
th e i n i t i a l
c o o rd in a te s are ta ke n as c o in c id e n t a l w it h the a c tu a l in s ta n ta n e o u s
c o o rd in a te s in re g io n 2, we see fro m the r e la t io n s
(14) t h a t the
t io n s , δ /δ x , δ /δ y , δ /δ z are e q u iv a le n t r e s p e c t iv e ly to
∂/∂a, ∂/∂b, ∂/∂c
1. Hadamar d , Sur l a P ro p o g a tio n des Ondes, 1903 p p .225-229.
opera­
on
-2 3 th e s u rfa c e o f d is c o n t in u it y .
Then f o r a second o rd e r d e s c o n tin u ity
e q u a tio n s (43) g iv e , s in c e ρ = ρ 0,
(44)
S u b s t it u t in g the v a lu e s o f the v a r ia t io n s in the l e f t members from
e q u a tio n s (38) and th e v a lu e s o f th o se in the r i g h t hand members from
( 4 1 ) , we o b ta in :
(45)
Since θ is n o t eq u a l to z e ro , we may d iv id e one e q u a tio n by a n o th e r and we
o b ta in :
λ :μ
: ν = α : β : γ
(46)
T h is shows t h a t the c h a r a c t e r is t ic segment is c o in c id e n t w ith the norm al
to th e s u rfa c e and hence we see t h a t the v a r ia t io n o f a c c e le r a tio n a t
the d is c o n t in u it y is a v e c to r p e rp e n d ic u la r to th e s u rfa c e .
q u a n tity λ α +μ
β
+ νγ
by i t s
The
form d e s ig n a te s th e p r o je c t io n o f the
c h a r a c t e r is t ic segment upon the no rm al b u t, s in c e th e c h a r a c t e r is t ic
segmen t is c o in c id e n t w ith th e n o rm a l, i t
fo rm e r.
is the a b s o lu te le n g th o f th e
I f th e r i g h t hand members o f th e above e q u a tio n s we have t h is
q u a n tity m u lt ip lie d by
, β
α
, γ
, w hich g iv e s th e p r o je c tio n s o f the
c h a r a c t e r is t ic segment upon th e c o o rd in a te axes, t h a t is
r e s p e c t iv e ly .
λ, μ,
ν
The th re e e q u a tio n s tha n reduce t o :
(47)
The v e lo c it y o f p ro p o g a tio n r e l a t i v e
to th e medium in i t s
a c tu a l s ta te
i s a c c o rd in g ly t h a t o f o r d in a r y sound o f i n f i n i t e s i m a l a m p litu d e 1 . I f
th e p re s s u re p, is n o t o n ly a f u n c tio n o f ρ b u t a ls o o f a , b , c in the
e q u a tio n s o f m o tio n used above we must have in th e r i g h t members
Lamb, Hydrodynam ics, 1911, p 455.
-2 4 ((1/ρ)(δp/δx)), ((1/ρ)(δp/δy)), ((1/ρ)(δp/δz)) r e s p e c t iv e ly .
S ince no d i s t i n c t i o n is in v o lv e d
b e tw e e n the c o o rd in a te axes we need o n ly c o n s id e r one o f the se ex­
p re s s io n s and r e s u lt s f o r th e o th e rs may be w r i t t e n down by c y c lic
change o f c o o rd in a te s .
U sing as i n i t i a l c o o rd in a te s th e a c tu a l in s t a n t ­
aneous c o o rd in a te s in re g io n 2, we have:
and
s in c e
∂p/∂ρ and ∂ p /∂ a are c o n tin u o u s a t th e d is c o n t in u it y .
On s u b s t it u t io n
o f t h is r e s u lt , the o n ly change a ffe c te d is a d isp la ce m e n t ofφ
'(ρ)
by
∂p/∂ρ.
Thus, even under the se c o n d itio n s
a c c e le r a tio n
, th e v a r ia tio n ;,
in
is norm al to th e s u rfa c e and th e v e l o c i t y o f p ro p o g a tio n
√(∂p/∂ρ).
We may g e n e ra liz e to a d is c o n t in u it y o f th e n ’ th o rd e r1 .
D i f f e r e n t ia t in g the e q u a tio n o f m o tio n w it h re s p e c t to t in th e i n i t i a l
system o f c o o rd in a te s n -2 tim e s and ta k in g v a r ia t io n s we o b ta in , i f the
i n i t i a l c o o rd in a te s a re th e in s ta n ta n e o u s c o o rd in a te s in re g io n 2,
(4 8 )
l e t us c o n s id e r o n e *o f th e q u a n t it ie s whose v a r ia t io n occu rs in th e r i g h t
member o f the above e q u a tio n s . R e s u lts f o r th e o th e rs may th e n be
deduced by c y c lic change o f c o o rd in a te s .
Thus
tThere A is an e x p re s s io n c o n ta in in g ^ d e r iv a tiv e s o f" ^
o f o rd e r le s s
tha n n -1 and d e r iv a tiv e s o f x , y , z o f o rd e r le s s th a n n .
From the la s t
e x p re s s io n we may tra n s fo rm to th e f o llo w in g e x p re s s io n :
Where B is
o f o ra e r le s s th a n n -1 i n
This was suggested by hadamard, lo c , e i t . ,
P0 and o f o rd e r le s s
b u t done here f a r th e f i r s t tim e *
-2 5 than n in x , y , z .
A t th e d is c o n t in u it y δ
/x
is e q u iv a le n t in o p e ra tio n
to ∂/∂a and in ta k in g v a r ia t io n s B is c o n tin u o u s and hence d is a p p e a rs .
Thus we o b ta in :
S im ila r ly , by c y c lic change o f c o o rd in a te s we have:
S u b s t it u t in g these e q u a tio n s in the e q u a tio n s (48) we o b ta in :
S u b s t it u t in g
le ft
the v a lu e s p r e v io u s ly fou nd f o r th e v a r ia t io n s in the
(38) and the r i g h t (41) members o f the e q u a tio n s , we o b ta in :
(49)
For θ n o t eq ua l to z e ro , we may d iv id e each o f the se e q u a tio n s by
(-θ )η
2
-
and th e e q u a tio n s become id e n t i c a l w it h the e q u a tio n s fou nd f o r
a second o rd e r d is c o n t in u it y .
The c o n c lu s io n s th e re drawn, th a t the
c h a r a c t e r is t ic segment is no rm al to th e s u rfa c e and t h a t th e v e lo c it y
o f p ro p a g a tio n is
√(φ'(ρ0)) th e r e fo r e s t i l l h o ld f o r a n ' t h o rd e r d is c o n ­
tin u ity .
3 . F i r s t O rder D is c o n tin u it y
a
. C o n tin u ity R e la tio n 1
l e t us now c o n s id e r a d is c o n t in u it y o f th e f i r s t o rd e r. L e t
Riemann, Uber d ie 1 o rtp flc ja z u n g ebener lu f t w e ll e n von e n d lic h e r SchwiBgangsw e ite - G o e ttin g e n Abhandlung t V I I I , 1860
-2 6 -
the v e lo c it y components, th e d e n s ity and th e p re s s u re
v a lu e s u 1
, v1, w1
, ρ1, p1 and u 2
, v2 , w
2
take on the
, ρ
,p2 , on the s u rfa c e o f d is c o n ­
2
t i n u i t y as we approach t h is s u rfa c e fro m th e re g io n s 1 and 2 r e s p e c tiv e ­
l y and l e t us choose th e x - a x i s c o in c id e n t w ith th e norm al to the
s u rfa c e and reckoned p o s it iv e i n the d ir e c t io n from the re g io n 1 to
the re g io n 2 .
L e t θ be th e v e lo c it y o f p ro p a g a tio n w ith re s p e c t to
any i n i t i a l s ta te in w hich the d e n s ity has the v a lu e ρ0 .
Then we have
from (41) and (3 8 ), s in c e α = 1, n = 1,
(50)
(51)
B lim in a t in g A
(52)
Ifθ
1
and θ2 are th e v e lo c it ie s o f p ro p a g a tio n w ith re s p e c t to the
re g io n s 1 and 2 reckoned as i n i t i a l s ta te s , we may s u b s t it u t e θ 1 a n d θ 2
f o r θ and ρ1 and ρ2 r e s p e c t iv e ly f o r ρ .
We thus see fro m th e above
e q u a tio n th a t the f o llo w in g r e l a t i o n must e x is t between these q u a l it ie s :
(53)
b . Dynam ical E q u a tio n and V e lo c it y Form ula
•
C o nside r a c y lin d e r o f i n f i n i t e s i m a l cro ss s e c tio n are a S
•T
norm al to the s u rfa c e o f d i s c o n t in u it y .
In a tim e d t a mass o f gasρ
t
d
S
θ
0
flo w s in t h is c y lin d e r acro ss th e s u rfa c e o f id is c o n t in u it y . T h is mass
o f gas may be co n sid e re d to be en clo se d by p is to n s in t h is tu b e .
D u rin g the passage o f t h is gas a cro ss th e s u rfa c e o f d is c o n t in u it y ,
one p is to n is on one s id e o f th e s u rfa c e and th e o th e r p is to n on th e
o th e r s id e .
Iffi,d t be ta k e n n e g lig ib le
in com parison w ith S, we may
n e g le c t the e f f e c t o f the p re s s u re on th e gas a cro ss the l a t e r a l w a ll
Riemann. lo c . c i t . .
-2 7 -
o f the c y lin d e r .
The fo rc e th e n a c t in g on t h i s gas is
w hich a c ts norm al to th e s u rfa c e .
A c c o rd in g to dyn am ical p r in c ip le s
the v a r ia t io n in momentum a t th e d is c o n t in u it y must he c o in c id e n t
w ith th e d ir e c t io n o f t h is fo rc e and hence th e v a r ia t io n o f momentum
and th e r e fo r e th e v a r ia t io n o f v e lo c it y ,
fa c e .
is a v e c to r norm al to the s u r­
We o b ta in the dynam ical e q u a tio n by e q u a tin g th e im pulse o f the
fo rc e to th e change in momentum,as f o llo w s :
o r d ro p p in g th e f a c t o r S d t ,
(54)
Both th e c o n t in u it y e q u a tio n and th e dyn am ical e q u a tio n were f i r s t
g iv e n by Riemann.
He a p p lie d them s p e c i f i c a l l y to p la n e waves.
By
e lim in a t in g u 1 - u 2between th e se two e q u a tio n s we o b ta in th e fo rm u la
f o r th e v e lo c it y o f p ro p o g a tio n :
(55)
As ρ1 approaches ρ
2
and p 1 approaches
p2
t h is e x p re s s io n approaches
the v a lu e : θ = ρ 1 /ρ 0
o r from e q u a tio n √(dp/dρ)
(53),θ
= d p /d ρ = θ 2
1
(56)
and hence the v e l o c i t y o f p ro p o g a tio n w ith re s p e c t to the medium on
e it h e r s id e o f th e d is c o n t in u it y approaches th e v e lo c it y o f p ro p o g a tio n
o f o r d in a r y sound o f i n f i n i t e s i m a l a m p litu d e
.
If
, however,
we assume P o is s o n ’ s a d ia b a tic law ( 21) we have:
and f o r g iv e n va lu e s^a s la rg e as we p le a s e by ta k in g p 2 s u f f i c i e n t l y
[of p1 and ρ1, θ becomes]
la r g e .
c . E q u a tio n o f energy
lo r d Rayleigh-*- o b je c te d to t h is s o lu t io n f o r p la n e waves
because the e q u a tio n o f energy is n o t s a t i s f i e d . .
L. R a y le ig h , Theory o f Sound - 1896, p p ,32 -3 2 , 40, 41.
By im p re s s in g upon
-2 8 -
the gas as a whole a v e lo c it y eq ua l and o p p o s ite to th e v e lo c it y o f
p ro p a g a tio n o f the d is c o n t in u it y , th e l a t t e r is b ro u g h t to r e s t w ith
re s p e c t to our f ix e d asses.
The phenomenon is th e n one o f ste a d y
m otio n and B e r n o u lli’s e q u a tio n o f energy (6) is a p p lic a b le .
T his g iv e s :
When th e d is c o n t in u it y is b ro u g h t to r e s t , u 1 = - θ1, u 2 = - θ2, and
from e q u a tio n (53) we o b ta in :
(57)
B e r n o u llis e q u a tio n o f energy mayAbe w r i t t e n as:
(5 8 }
Here u(, f>, and p f may be c o n s id e re d as c o n s ta n t and u ^ , & , p z as
v a r ia b le .
Then d i f f e r e n t i a t i n g we have:
and in t e g r a t in g we o b ta in :
(59)
T his is the o n ly law o f p re s s u re f o r w hich the energy c o n d itio n and th e
e q u a tio n o f c o n t in u it y a re s a t i s f i e d .
T his is n o t f u l f i l l e d
f o r any
a c tu a l gas and on t h is b a s is R a y le ig h con clu de s t h a t a f i r s t o rd e r
d is c o n t in u it y i s im p o s s ib le w ith o u t d is s ip a t iv e fo r c e s .
A fo rc e
a c tin g on th e gas is a new q u a n t it y e n te r in g the d yn a m ica l e o u a tio n and
may be de term in ed so as to produce a ste a d y s ta te w ith any la w o f
p re s s u re .
Weber'*' c o n s id e rs th e energy change in a d is c o n t in u it y o f
the f i r s t o rd e r ab i n i t i o
and f in d s f o r th e a d ia b a tic law o f P o isson (21)
r
a change o f energy i n p a s s in g the d is c o n t in u it y .
He says t h a t we must
in t h is case c o n s id e r the lav/ o f C a rn o t, in accordance w ith w hich th e re
may be a lo s s o f energy b u t ne ver a g a in .
Prom t h i s c r i t e r i o n he
1. Weber, D ie P a r t ie lle n D if f 'e r e n t ia l- g le ic h u iz e n , Y o l . I I 1901, p p .489-498.
-2 9 -
determ ines t h a t a d is c o n t in u it y o f c o n d e n sa tio n is p o s s ib le b u t one o f
r a r e f a c t io n im p o s s ib le .
Weber says t h a t th e lo s s o f energy must be made
up by the form es th a t move th e p is to n s .
does n o t go in h is e x p la n a tio n .
Beyond
t h is s ta te m e n t he
I t h in k a s o lu t io n to t h i s q u e s tio n
must be o b ta in a b le by a c o n s id e r a tio n s o le ly o f what happens in th e
im m ediate ne ighborhood o f th e d is c o n t in u it y and t h a t , as R a y le ig h
says, f o r th e a d ia b a tic law o f Pois s o n th e d is c o n t in u it y does n o t obey
th e law o f energy and t h a t t h is s o lu t io n is th e r e fo r e in v a l i d ,
d. H u g o n io t's la w .
H u g o n io t1 o b je c te d to th e use o f the a d ia b a tic law o f
P oisson (21) as
the v a l i d i t y o f t h is law depends on
w hich is c o n tin u o u s .
He s a id th a t t h is law is no more v a l id f o r a b ru p t
d is c o n tin u o u s changes o f d e n s ity .
the l a t t e r case.
a change o f d e n s ity
He d e riv e s a law w h ich i s v a lid f o r
He uses th e changeofi n t e rn a l energy o f a gas a
s
the energy
change n o t accounted f o r by th e k in e t ic energy andth
y e x te r n a l
b
n
rkd
o
w
e
fo rc e s b u t w h ich must e x is t because o f the law o f c o n s e rv a tio n o f e n e rg y.
The in t e r n a l energy o f a mass o f gas i s a f u n c tio n s o le ly o f i t s
s ic a l s ta te and s in c e a l l energy is a d d itiv e i t
gas.
L e t us c o n s id e r a g a in the mass o f gas,
phy­
v a rie s as th e mass o f
ρ0θsd t, w h ich flo w s acro ss
the s u rfa c e o f d is c o n t in u it y th ro u g h a c y lin d e r o f i n f i n i t e s i m a l c ro s s s e c tio n S in th e tim e d t .
Then th e in t e r n a l energy o f t h i s gas s u f­
fe r s a change o f m agnitude [η(ρ1,p1) - η(ρ2,p2)]ρ0θsdt where η (ρ ,p ) is the
s p e c if ic in t e r n a l e n e rg y, t h a t i s ,
th e in t e r n a l energy £ e r u n i t mass.
Since the work done by th e -pressures in i t s
passage from one s id e o f th e
s u rfa c e o f d is c o n t in u it y to the o th e r i s ( p ( u ( - p ^ u J S d t and the change
in k in e t ic energy is J _ p eQ S d t ( u * - u * ) , th e e q u a tio n o f energy beX
comes, d ro p p in g th e f a c t o r S d t ,
(60)
H ugoniot,, J o u rn a l de l ’ eeole P o ly te c h n iq u e , 1887, 1889
- 30 -
T his e q u a tio n should rem ain v a lid
if
any v e lo c it y is added to the gas
as a whole and the above e q u a tio n s h o u ld , t h e r e fo r e , c o n ta in o n ly th e
q u a n tity u 1 - u 2. To make t h is e v id e n t, m u lt ip ly the dyn am ical e q u a tio n
(54) by (u 1 + u 2)/2) and s u b tr a c t fro m the p re c e e d in g e q u a tio n and we o b ta in :
(61)
E lim in a tin g u 1 - u 2 by means o f th e e q u a tio n o f c o n t in u it y (5 2 ), we o b ta in
(62)
For a p e r f e c t gas the in t e r n a l energy has th e fo rm 1 η (ρ1p) =
(1/(y-1))(p/ρ)
and in t h is case we o b ta in :
(63)
or
(64)
T his is th e a d ia b a tic law o f H u g o n io t the v a l i d i t y o f w h ich is r e ­
cognized w ith o u t q u e s tio n by a l l the French s tu d e n ts o f t h is q u e s tio n .
H ugoniot c a lle d t h is law th e d yn am ical a d ia b a tic law in c o n tr a s t w ith
P o is s o n 's r e l a t io n w hich he c a lle d th e s t a t i c a d ia b a tic la w .
approaches
1
ρ
2
ρ
form :
As
and p 1 approaches p 2 , H u g o n io t's law approaches the
(dp/dρ) = (yp/ρ)
and hence agrees w ith the law o f P o is s o n .
By the v e ry method o f i t s
d e r iv a tio n the la w o f H u g o n io t s a t i s f i e s th e c o n d itio n o f energy and
hence is n o t open to the o b je c t i o n w h ich we fou nd in th e a p p lic a t io n
o f P o is s o n 's la w to a f i r s t o rd e r d is c o n t in u it y
e. En tro p y and Thermodynamic P o t e n t ia l
We s h a ll need th e q u a n t it ie s , e n tro p y and thermodynamic
p o t e n t ia l f o r l a t e r d is c u s s io n and i t
is a d v is a b le here to d e fin e them.
1. Sur le P ro p o g a tio n des Ondes * Hadamard, 1903, p 191
-3 1 -
I f a system o r p a r t o f a system a t an a b s o lu te te m p e ra tu re T re c e iv e s
a q u a n tity o f h e a t dQ, the in c re a s e o f e n tro p y 12 i s :
(65)
I f th e te m p e ra tu re o f th e system i s n o t c o n s ta n t, we must b re a k i t up
in t o elem ents and sum up th e e n tro p y o f each e le m e n t.
How the change
in th e in t e r n a l energy in ch a n g in g fro m one s ta te to a n o th e r is made
up o f th e h e a t g a in and o f the w ork done on th e gas, when the work
n e c e s s a ry to produce th e change in k in e t ic
ed from c o n s id e r a tio n .
We s h a ll l e t
^
energy as a whole is a b s t r a c t ­
be the s p e c if ic in t e r n a l e n e rg y,
and we s h a ll re cko n th e w o rk, W, as p o s it iv e when i t
the gas and n e g a tiv e i f
it
i s done on the gas.
is work done by
Then we must have, f o r
a u n i t mass o f gas
d Yj = dQ - dW
S u b s t it u t in g th e va lu e o f dC fro m ( 6 5 ) we o b ta in :
(66)
From the d e f i n i t i o n o f w ork we have:
dW=pdv, where v is th e s p e c if ic volume o f th e gas
S u b s t it u t in g t h is
va lu e o f dW in
(66) we o b ta in :
(67)
The therm odynam ical p o t e n t ia l is d e fin e d
2
as :
(68)
D i f f e r e n t ia t in g we have:
S u b s t it u t in g th e va lu e o f d »-j from (67) we o b ta in :
1. G. H. B ryan, Thermodynamics, 1107, p . 58
2. B ryan, Thermodynamics, 1907, p p .9 1 ,9 2 .
-3 2 -
Hence we have:
(69)
(70)
F or ir r e v e r s ib le processes th e e n tro p y o f a system must in c re a s e .
In f a c t the i n crease o f e n tro p y may be ta ke n as a measure o f the
i r r e v e r s i b i l i t y o f a p ro c e s s .
A r e v e r s ib le pro cess is o n ly an id e a l
and we may c o n s id e r th e in c re a s e o f e n te ro p y a ne cessa ry c o n d itio n in
a l l a c tu a l p h y s ic a l changes.
f . A p p lic a tio n o f the E n tro p y C o n d itio n to a F i r s t O rder D is c o n t in u it y .
Duhem1
uses as a fo u n d a tio n o f h is s tu d y o f f i r s t o rd e r
d is c o n t in u it ie s e q u a tio n s (6 8 ),
(52) and ( 54 ) .
(69)
(70)
and the e q u a tio n s o f Riemann
The c o n d itio n o f in c re a s e o f e n tro p y , he wr i t e s by means
o f (6 9 ):
(71)
He a ls o uses th e c o n d itio n t h a t the p re s s u re must in c re a s e w ith in c re a s e
o f d e n s ity o r t h a t d
/ρ
p
is p o s it iv e a lw a ys.
By means o f e q u a tio n (70)
t h is e q u a tio n may be w r it t e n :
(72)
He a ls o uses H u g o n io t's law in th e g e n e ra l fo rm
(6 2 ).
are
The d is c o n t in u it y is
c o n s id e re d fu n c tio n s o f ρ
and T.
A l l fu n c tio n s
supposed to have v a lu e s o f th e d e n s ity and te m p e ra tu re ρ 1 ,T 1 by
approach from th e re g io n 1 and ρ 2 ,T2 b y approach from the re g io n 2.
A t the d is c o n t in u it y ρ1 and T (
are the n fu n c tio n s o f Px .
a re c o n s id e re d as f ix e d and a l l fu n c tio n s
We may th u s w r it e :
S u b s titu te in e q u a tio n (62) th e v a lu e o f b from (68) and we o b ta in :
(73)
D i f f e r e n t ia t in g w ith re s p e c t to
we o b ta in :
and u s in g e q u a tio n s (69) and (70)
(74)
. Duhem, Z e i t s c h r i f t f u r P h y s ik a lis c h e Chemie 69, 1909, pp. 169-186.
-3 3 -
By d i f f e r e n t i a t i n g the v a lu e o f σ (ρ2,T2) = ∑ ( ρ2) from e q u a tio n (69)
and the va lu e o f ρ2 = Π ( ρ 2 ) from e q u a tio n (70) w ith re s p e c t to ρ2 we
o b ta in :
(75)
(76)
I t is o f advantage to in tro d u c e here th e q u a n t it y V ( ρ2,T2) = ^ ^-TRpJj
w hich is th e o r d in a r y v e l o c i t y o f sound o f i n f i n i t e s i m a l a m p litu d e in
the re g io n 2 w it h re s p e c t to t h i s medium.
q u a n tity
(dp2/dT2)(p2
U sin g t h is q u a n t it y and the
c o n s ta n t) we may express th e r e s u lt o f the e lim in a t io n
o f dθ(ρ2)/dρ2 between the above two e q u a tio n s in th e f o llo w in g fo rm :
(77)
In e q u a tio n (74) s u b s t it u t e :
o b ta in e d from th e v e lo c it y e q u a tio n o f Reimann (5 5 ).
s u lt in g e q u a tio n and e q u a tio n (7 7 ), e lim in a te
Between th e r e ­
dΠ(ρ2)/dρ2and we o b ta in :
(78)
Duhem1
w r it e s t h is e q u a tio n w ith th e r i g h t member o f o p p o s ite s ig n
b u t h is er r o r is e v id e n t on com plete in v e s t ig a t io n .
him to advance some wrong
T h is e r r o r
le d
c o n c lu s io n s b u t th e y are e a s ily r ig h t e d .
T his e r r o r is e v id e n t a ls o fro m th e c o n t r a d ic t io n in the c o n c lu s io n s
w hich he draws in the g e n e ra l case and in th e case o f a p e r f e c t gas.
From t h is e q u a tio n we see t h a t f o r ρ2 =ρ1
Duhem2
does n o t m e n tio n t h a t t h i s c o n c lu s io n m ig h t have been
from e q u a tio n (71) s in c e Π ( ρ 2 ) = P1 , whenρ
2
= ρ
. D iffe r e n tia tin g
1
e q u a tio n (78) we o b ta in an e q u a tio n o f the f o llo w in g form*.
L. Loc. c i t . p.
1. Loc. c i t . p .
o b ta in e d
-3 4 -
(80)
where A and B a re f i n i t e
fo r ρ 1 = ρ 2 .
Then f o r ρ1 = ρ2 we must have from
the above e q u a tio n ,
(81)
Thus f o r a d is c o n t in u it y w it h an i n f i n i t e s i m a l v a r ia t io n in d e n s ity ,
the v a r ia t io n i n e n tro p y is an i n f i n i t e s i m a l o f a t le a s t th e t h i r d o rd e r.
Since
ρ1θ1 = ρ2θ2(see e q u a tio n (53)
)
θ1
and θ2
a re o f the same s ig n .
A c c o rd in g to th e c o n v e n tio n o f s ig n a lre a d y used, i f θ 1 a n d θ 2
a re
p o s it iv e , the gas passes from th e re g io n 2 to the re g io n 1 and
∑ (ρ2)-σ(ρ1,Τ1)
and θ2
must he n e g a tiv e s in c e th e e n tro p y must in c re a s e ;
ifθ
1
are n e g a tiv e , th e gas passes from the re g io n 1 to th e re g io n 2
and ∑(ρ2)-σ(ρ1,Τ1) must be p o s it iv e .
Thus th e d iffe r e n c e in e n tro p y ,
∑(ρ2)-σ(ρ1,Τ1) is o p p o s ite in s ig n to θ1 and θ 2 . S ince th e v a r ia t io n
o f e n tro p y is o f th e t h i r d o rd e r ∑(ρ2)-σ(ρ1,Τ1) has th e s ig n o f (ρ2-ρ1)[(d3∑
(ρ2)/dρ23]ρ2=ρ1
as lo n g as ρ1 -
does n o t exceed in a b s o lu te v a lu e a c e r t a in l i m i t .
2
ρ
By d i f f e r e n t i a t i n g e q u a tio n (80) and p la c in g ρ2 = ρ1 we e a s ily o b ta in :
(82)
For b r e v it y l e t us in tro d u c e the f u n c tio n
(83)
Then ∑(ρ2)-σ(ρ1,Τ1)
has th e s ig n o f (ρ 2 - ρ 1 ) H ( ρ1,Τ1)
s ig n s o p p o s ite to t h i s q u a n t it y .
a p o s it iv e d is c o n t in u it y
and θ1 and θ2 must have
In gases f o r w h ich H ( f,,T, ) is p o s it iv e
(6>tf) can o n ly he p ro po ga te d i f
p, >
or i f
the wave is one o f co n d e n s a tio n and f o r a gas f o r w hich H (^?,T, ) is
n e g a tiv e , a p o s it iv e d is c o n t in u it y can o n ly be p ro p o g a te d i f
i f the wave is one o f r a r e f a c t io n .
P, < p x o r
In th e same way, o f c o u rs e , i f
the
- 35 -
d is c o n t in u it y is p ro p o g a te d in th e n e g a tiv e d ir e c t io n , co n d e n s a tio n o r
r a r e f a c t io n are p o s s ib le a c c o rd in g as H ( ρ1, T 1 ) is p o s it iv e o r n e g a tiv e
r e s p e c tiv e ly .
For a c tu a l gases H (ρ1,T1) is p o s it iv e and hence o n ly
con de nsa tion s a re p o s s ib le .
E q u a tio n (78)
in v o lv e s the q u a n t it y V ( ρ2T2) - θ 2 2 . By
d e te rm in in g the s ig n o f t h i s q u a n t it y , we d e te rm in e th e r e l a t i v e m agni­
tude o f the v e lo c it y o f p ro p a g a tio n , w it h re s p e c t to th e gas i n re g io n
2 compared w ith the v e lo c it y o f p ro p a g a tio n o r o r d in a r y sound in t h i s
re g io n . In t h is e q u a tio n th e c o e f f i c ie n t o f
dΣ(ρ2)/dρ2 i s p o s it iv e
if
i n a b s o lu te v a lu e is l e ss th a n a c e r t a in l i m i t s in c e 2ρ1ρ2θ(ρ2) is
p o s it iv e .
A ls o
dΣ(ρ2)/dρ2
has the s ig n o f H (ρ1,T1) and th e r e fo r e is
p o s it iv e o r n e g a tiv e ac c o rd in g as th e wave is
r a r e f a c t io n .
one o f c o n d e n s a tio n o r
We may express the se r e s u lt s by s a y in g t h a t
dΣ(ρ2)/dρ2 and
ρ2-ρ1 have th e same s ig n i f th e re g io n 2 is b e h in d th e wave and o p p o s ite
s ig n s i f
t h is re g io n is in f r o n t o f t h i s wave.
From the above m entioned
e q u a tio n we th u s see th a t w it h re s p e c t to th e re g io n behind the wave,
th e v e lo c it y o f p ro p a g a tio n is le s s th a n t h a t o f o r d in a r y sound and w it h
re s p e c t to th e re g io n i n f r o n t o f th e wave i t
o r d in a r y sound.
is g re a te r th a n th a t o f
These r e s u lt s h o ld , o f co u rs e , p r o v i d e d ρ
does n o t
-1
2
exceed a c e r t a in l i m i t in a b s o lu te v a lu e .
Duhem1 a ls o examines th e s ig n o f the q u a n t it y θ 2 2 - [V ( ρ 1 ,T 1 ) ]2 . In so
d o in g he compares the v e lo c it y o f p ro p a g a tio n w ith re s p e c t to one re g io n
w ith th e v e lo c it y o f o rd in a r y sound in th e o th e r medium.
q u a n tity is
z e ro .
For ρ 2 = ρ 1 t h is
By d i f f e r e n t i a t i n g th e e x p re s s io n f o r the v e lo c it y
as g iv e n by Riemann (55) we o b ta in :
(84)
where β
-1
)=
2
(ρ
For
ρ 2 = ρ 1 , β (ρ 2 )= o
.. lo c . c i t . p .
dΠ(ρ2)/dρ2 - (2ρ2-ρ1)[π(ρ2)-p1]
th e r e fo r e we must examine
<*3j o j . By d i f f e r e n t -
-3 6 -
i a t i n g we have:
(85)
For ρ 2 = ρ 1
t h is
is
zero and we must examine th e n e x t d e r iv a t iv e .
F o r t h is we o b ta in th e e x p re s s io n :
(86)
and
say
Thus we see t h a t f o r
β ( ρ2) has i t s
(87)
le s s th a n a c e r t a in l i m i t in a b s o lu te v a lu e -
s ig n o p p o s ite to t h a t o f K (ρ 1 T1)and θ22- [V(ρ1T1)]2 has the
s ig n o f - (ρ2 - ρ1)K (ρ l Tl)
I f K ( on T(J ls p o s it iv e , th e v e lo c it y o f p ro p a g a tio n o f th e d is c o n t in u it y
w ith re s p e c t t o th e le s s dense re g io n i s g re a te r th a n the v e lo c it y o f
sound in the more dense re g io n ;
if
is n e g a tiv e , th e v e lo c it y o f
p ro p a g a tio n o f th e d is c o n t in u it y w it h th s p e c t to th e more dense re g io n
is le s s th a n the v e lo c it y o f sound in th e le s s dense re g io n .
Duhem'1" the n a p p lie s these th e o r ie s to a p e r f e c t gas.
I f C and c are
the s p e c if ic h e a ts u n de r c o n s ta n t p re s s u re and volume r e s p e c t iv e ly we
have iiu g o n io t's law (64) in th e f o llo w in g fo rm :
(88)
E it h e r th e n u m era to r o r d e no m in ator o f the f r a c t io n in th e r i g h t
member a te p o s it iv e and s in c e the p re s s u re s must be p o s it iv e we must
have b o th n u m era to r and de no m in ator p o s it iv e .
(89)
We may w ith o u t lo s s o f g e n e r a lit y a t t r i b u t e the s u b s c r ip t 2 to th e
g re a te r o f th e two d e n s it ie s .
o f its e lf.
Then th e f i r s t s n e g u a lity is f u l f i l l e d
The second may be w r i t t e n in th e
fo rm :
(90)
Loc. c i t . p
-3 7 -
For most d ia to m ic gases C
v a lu e n e a r 6.
p2
/ c
.4 a p p ro x im a te ly and ρ 2 /ρ 1 has a l i m i t i n g
1
=
As the d e n s ity in c re a s e s w ith a f i x e d ρ1 the p re s s u re
becomes i n f i n i t e
as ρ 2 approaches
(C + c)/(C-c)ρ1.
E lim in a t in g p 2 between
e q u a tio n s (55) and (88)we o b ta in , when we p la c e $ = θ1, and ρ 0 = ρ1
and
Then we have :
(91)
T h is shows t h a t th e v e lo c it y o f p ro p a g a tio n o f a d is c o n t in u it y w it h r e ­
spe ct to the re g io n be hind the wave is le s s tha n th e v e lo c it y o f sound
in th e same re g io n and the v e lo c it y w ith re s p e c t to th e r e g io n in f r o nt
o f the wave is g re a te r th a n th a t o f sound in t h is r e g io n .
the fu n c tio n
o f e q u a tio n
We see t h a t
( 87), K(ρ1,T1),has th e f o llo w in g fo rm :
(92)
T his i s p o s it iv e o r n e g a tiv e a c c o rd in g as C/c is le s s o r g re a te r th a n 3
I n a l l known cases the fo rm e r is t r u e .
shown to he p o s it iv e f o r ρ
/2
1
In t h is case θ22-[V(ρ1,T1)]2
le s s th a n 2c/(C-c)
and n e g a tiv e i f
it
is e a s ily
is
g re a te r th a n t h i s q u a n t it y .
M. E. Jouguet^ o b ta in e d some o f the above r e s u lt s p re v io u s
to M. huhem b u t by analagous m ethods.
e n tro p y and d e riv e d the id e n t ic a l
He a ls o used th e d r i t e r i o n o f
o n d itio n s to those g iv e n above f o r
the p o s s i b i l i t y o f a c o n d e n sa tio n and r a r e f a c t io n . He shows th a t i f the
s
medium i f o f s m a ll v is c o s it y , a s m a ll s tra tu m o f g re a t change in v e lo c it y
and d e n s ity w i l l r e s u lt such t h a t the v is c o s it y may be n e g le c te d e v e ry »d
where save in t h i s s tra tu m .
I f th e th ic k n e s s o f t h is s tra tu m is so
Jouguet, Comptes Hendue 138 p 1685, 1904; Comptes Hendus 139, p .7 8 6 , 1904
-3 8 -
i n f i n i t e l y s m a ll, th e laws o f p ro p a g a tio n are tho se g iv e n by the fo rm ­
u la e o f Riemann (e qu a tio n s 52 and 54) and o f Hu g o n io t (e q u a tio n 6 4 ) .
g . M o tio n F o llo w in g an I n i t i a l P lane Wave D is c o n t in u it y .
An i n i t i a l d is c o n t in u it y o f th e f i r s t o rd e r w i l l n o t in
g e n e ra l, be p ro pa ga te d a s such f o r th e e q u a tio n s o f Riemann,
(5 4 ), to g e th e r w ith (18)
, g iv in g th e law o f p re s s u re , w i l l n o t, in gen­
e r a l be s a t is f ie d f o r a r b i t r a r y va lu e s o f u 1
, ρ1,
d is c o n t in u it y .
(52) and
u 2 ,ρ 2 in an i n i t i a l
The i n i t i a l d is c o n t in u it y may re s o lv e i t s e l f in t o
two d is c o n t i n u it ie s o f th e f i r s t o rd e r o r in t o d is c o n t in u it ie s o f the
second o rd e r.
(1) R iem ann's Treatm ent
Riemann1
tr e a te d o f an i n i t i a l p la n e wave d is c o n t in u it y
in an unbounded gas in w h ich the re g io n s 1 and 2 were in a c o n s ta n t
u n ifo rm c o n d itio n o f v e lo c it y and d e n s ity suchthat th ro u g h o u t re g io n s 1 and 2
andUi
f>, and fares pe-csirively
th e v e lo c it y was u, and the d e n s ity . . Weber g iv e s Riem ann's tre a tm e n t
i n a v e ry c o n c is e , c le a r manner.
Riemann showed,
1 s h a ll here f o llo w h is tre a tm e n t.
t h a t , co n fo rm a b ly w ith the e q u a tio n s o f m o tio n (3)
and (4) a p p lic a b le to re g io n s where a l l th e q u a n t it ie s in v o lv e d are
c o n tin u o u s and w ith e q u a tio n s (52) and (54) a p p lic a b le to a f i r s t
o rd e r d i s c o n t i n u it y , assum ing th e p re s s u re g iv e n by ( 1 8 ;, an i n i t i a l
p la n e wave d is c o n t in u it y , c o u ld be s a t i s f i e d by one o f fo u r cases de­
pending upon th e i n i t i a l v a lu e s o f the v e lo c it y and d e n s ity in re g io n s
1 and 2 .
These eases in v o lv e r e s o lu t io n in t o d is c o n t in u it ie s a f
com pression and a s p e c ia l ty p e o f r a r e f a c t io n wave.
a lre a d y been d is c u s s e d .
The fo rm e r have
The l a t t e r type o f wave in v o lv e s th e a p p l i­
c a tio n to a c e r t a in domain o f the x - t p la n e o f a s o lu t io n o f th e typ e
u
+ c = rF Jfyfijo) + *
T h is s o lu t io n may e a s ily be v e r i f i e d
. doc. c i t .
. '7eber , D ie f a r t i e l l e n D if f e r e n t ia l- g le ig h u n g e n IIp p 4 8 0 -8 8
-3 9 -
by s u b s t it u t io n in e q u a tio n s (3) and ( 4 ) .
1 s h a ll d is c u s s t h i s s o lu t io n
more f u l l y l a t e r h u t 1 s h a ll need the r e s u lt f o r use in the p re s e n t
case.
K iem ann's fo u r cases a re , th e n :
(1) a r e s o lu t io n in t o two
d is c o n t in u it ie s o f com pression pro p a g a te d in o p p o s ite d ir e c t io n s w it h
re s p e c t to th e gas,
(£) a r e s o lu t io n in t o two r a r e f a c t io n waves p ro ­
pagated in o p p o s ite d ir e c t io n s w ith re s p e c t to th e gas,
(3) a re s o ­
l u t i o n in to a p o s i t i v e l y p ro p a g a te d d is c o n t in u it y and a n e g a tiv e ly
porpagated r a r e f a c t io n wave, and (4) a r e s o lu t io n in t o a n e g a tiv e ly
propagated d is c o n t in u it y and a p o s i t i v e l y pro pa ga te d r a r e f a c t io n wave.
e may ro u g h ly see the c o n d itio n s f o r some o f these case s, l o r the f i r s t
case, the in te rm e d ia te re g io n c re a te d i s a re g io n o f g re a te r d e n s ity
th a n e it h e r o f the domains 1 and 2 and th e v e lo c it ie s must o b v io u s ly
be such th a t the two masses o f gas in re g io n s 1 and 2 t& jodcth'U jaiatl,
a n o th e r.
one
L e t us s tu d y the se cases in d e a t i l .
C o nside r K iem ann's f i r s t case.
The i n i t i a l d is c o n t in u it y
re s o lv e s i t s e l f in t o two d is c o n t in u it ie s o f com p ression .
In f r o n t o f
the f i r s t d is c o n t in u it y u and p r e t a in t h e i r i n i t i a l v a lu e s u t and
and be h in d th e second d is c o n t in u it y th e y r e t a in t h e i r i n i t i a l v a lu e s
u,>
w h ile in the re g io n between th e two d is c o n t in u it ie s th e y have
the c o n s ta n t v a lu e s u ^ p
. Then th e v e l o c i t i e s o f p ro p a g a tio n o f th e
two d is c o n t in u it ie s a re c o n s ta n ts .
We may re p re s e n t th e p o s it io n o f
each o f the d is c o n t in u it ie s a t a tim e t by a p o in t in a p ane in w hich
x and t are ta ke n as r e c ta n g u la r c o o rd in a te s .
In t h i s p la n e th e m o tio n
o f each d is c o n t in u it y is re p re s e n te d by a l i n e
s in c e the v e lo c it y o f
p ro p a g a tio n
i i s
c o n s ta n t.
x - a x is is such th a t i t s
The angle made by t h i s l i n e w it h th e
c o ta n g e n t is
dX ,th e v e l o c i t y o f p ro p a g a tio n ,
"2SF
In the accompanying h ig . 1 th e d is c o n t in u it ie s are pro p a g a te d a lo n g the
lin e s 1 and 2 w hich make a n g le s
o f e q u a tio n (34) we th e n o b ta in :
and <Xj_ w ith the x - a x is .
By means
-4 0 -
(93)
(94)
F ig . 1.
From the above fo rm u la e ,
(95)
(96)
A dding we have:
(97)
The fu n c tio n , ((ρ-ρ1)φ(ρ)-φ(ρ1))/ρρ1 = ( (1/ρ1)- (
1
/
ρ
)
) [φ(ρ)-φ(ρ1)]for
is an in c r e a s in g f u n c tio n o f a s in c e each f a c t o r is p o s it iv e and in c re a s e s
w it h in c re a s e o f ρwhenρ>ρ1.For Riem ann's f i r s t cas e, ρ' must b e g r e a te r tha n
b o th ρ1 andρ
2
and hence th e r i g h t member o f the above e q u a tio n i s an i n ­
c re a s in g f u n c t io n o f ρ ' .
When
'=ρ1 and when ρ ' =ρ 2 , the r i g h t member
ρ
o f the above e q u a tio n becomes ((ρ1-ρ2)[φ(ρ1)-φ(ρ2)])/ρ1ρ2
and th e r e fo r e we
must have the f o llo w in g in e q u a lit y :
(98)
I f t h is in e q u a lit y i s
fu lfille d ,
th e re is a s in g le v a lu e o f ρ' w hich
s a t is f ie s e q u a tio n (97) and when ρ ' is fo u n d , u'
o f e q u a tio n s (95) o r (9 6 ).
The v e l o c i t i e s
is o b ta in e d from e it h e r
o f p ro p a g a tio n are found
from (93) and (94) and th e n e ce ssa ry r e l a t io n ,
cot
is
α2> c o t α1
, o f i t s e l f , v e r ifie d .
I f the c r i t i c a l in e q u a lit y
(98)
(98) above is an
e q u a lity , we see from th e l a s t e q u a tio n t h a t ρ ' equals e it h e r ρ
1
o r ρ2 and
-4 1 we have a s in g le d is c o n t in u it y p ro p a g a ted backward o r fo rw a rd re s p e c t­
iv e ly .
C o nside r n e x t Riem ann's second case.
Fig . 2
we ta ke fo u r s t r a ig h t li n e s ,
In the accompanying
1 ,2 ,3 ,4 w ith th e a n gles α 1 ,α 2 ,α 3 ,α 4
w ith th e x a z is r e s p e c t iv e ly and assume in th e s e c to r ( - ∞ 0 1 ) u and
p c o n s ta n t and e q u a l to t h e i r i n i t i a l v a lu e s u 1,ρ1 and in the s e c to r
(4 , 0 +∞ ) lik e w is e c o n s ta n t and equ a l to u 2 ,ρ 2 .
l e t u and ρ
be c o n s ta n t and eq u a l to u ',ρ'.
In the s e c to r (2 ,0 ,3 )
In th e s e c to rs ( 1 , 0 , 2 )
and ( 3 ,0 ,4 ) we assume s o lu tio n s o f th e t y pe u =
±∫((√φ'(ρ))/ρ)dρ + c = ±√(φ'(ρ)) + (x/t)
ta ke n so th a t a lo n g the lin e s 1 , 2 ,3 ,4 these s o lu tio n s s h a ll b e con­
tin u o u s w ith the va lu e s in th e b o rd e rin g dom ains.
t y pe has the c h a r a c t e r is t ic t h a t f o r
s t a n t,
in o th e r w ords, u and ρ
x /t
A s o lu t io n o f t h i s
c o n s ta n t, u and ρ are c on­
have c o n s ta n t v a lu e s a lo n g e v e ry li n e
em anating from 0 .
Fi g . 2.
L e t us f o r b r e v it y w r it e f ( ρ ) = ∫((φ'(ρ ))/ρ)dρ.
We may ta b u la te the above
c o n d itio n s as f o llo w s :
(99)
The c o n d itio n o f c o n t in u it y between these re g io n s g iv e s th e f o llo w ­
in g r e l at io n s :
-4 2 -
(10 0)
From these e ig h t e q u a tio n s we must d e te rm in e the e ig h t q u a n t it ie s ,
c , c ' , u ' , ρ ', α 1 , α 2 , α 3 , α 4 .
E lim in a t in g c and c ' we o b ta in :
(101)
(102)
S u b tra c tin g we o b ta in :
(103)
S ince the waves are o f r a r e f a c t io n , ρ' must be s m a lle r th a n b o th
ρ1 and ρ 2 .
The d e r iv a t iv e fu n c tio n :
f '( ρ ) =
is p o s it iv e and f (ρ
(√φ'(ρ))/ρ
) th e r e fo r e decreases as ρ
d e cre a se s.
u 1 - u 2 , g iv e n by th e above e q u a tio n , is le s s than i t s
ρ' =
For ρ
,
2
>
1
v a lu e f o r
and we have:
2
ρ
(104)
For ρ2 > ρ1, u 1- u 2 is le s s th a n i t s
va lu e f o r ρ ' = ρ1 and we have:
(105)
I n o rd e r t h a t a vacuum he n o t c re a te d in th e gas, f o r c e r t a in laws o f
pre s s u re , we must f u r t h e r r e s t r i c t u 1 - u 2 to b e in a b s o lu te v a lu e le s s
than a l i m i t d e term in ed by th e law o f p re s s u re .
P o is s o n 's la w , φ(ρ)=aρ y
F o r exam ple, f o r
and f (ρ ) = (2a(√y))/(y-1))ρ(y-1)/2 w h ic h , s in c e y is
g re a te r th a n 1 and s in c e ρ
From e q u a tio n (103) t h is
is e s s e n t ia lly p o s it iv e , must be p o s it iv e .
in v o lv e s th e l i m i t a t i o n
u1 - u 2< - f (ρ 1 ) - f ( ρ 2 ).
For B o y le 's law no l i m i t a t i o n is ne ce ssa ry s in c e f ( ρ ) = a lo g ρ
be any v a lu e from - ∞
to +∞
.
can
I f th e l i m i t a t i o n ju s t m entioned and
the in e q u a lit y (104) o r (105) w h ich e ve r i s a p p lic a b le , is f u l f i l l e d ,
the n e q u a tio n (103) g iv e s a u n iq u e va lu e f o r ρ '.
Knowing ρ ' , from
e it h e r e q u a tio n (101) or (102) we may o b ta in u ' w hich l i e s between
-4 3-
u 1andu
2
and from e q u a tio n s (100) we may th e n o b ta in th e c o ta n g e n ts
o f th e a n g le s α 1 , α 2 , α 3,
and α 4 .
These a re in c o r r e c t o rd e r o f
m agnitude w ith o u t f u r t h e r r e s t r i c t i o n s .
C o nside r n e x t Ri emann's t h i r d case.
In t h i s case a d is c o n ­
t i n u i t y o f c o n d e n sa tio n is p ro p a g a te d fo rw a rd s and a wave o f r a r e f a c t io n
backward.
T his in v o lv e s th e assum ption t h a t ρ1 is g re a te r th a n ρ 2 .
In the x - t p la n e (see F ig . 3) we have th e wave o f r a r e f a c t io n in th e
s e c to r ( l , o , 2 ) and th e p o s it iv e ly p ro pa ga te d d is c o n t in u it y a lo n g th e
lin e
(o , 3)
.
F ig . 3.
W ith the te rm in o lo g y in d ic a te d in the accompanying d iag ram , we may
ta b u la te th e f o llo w in g c o n d itio n s :
( - ∞ , o , l ) : u = u 1 ρ = ρ 1 , c o n s ta n t
( l,o ,2 )
:
( 2 ,0 ,3 )
: u
u = - f ( ρ ) + c = (√φ'(ρ)) + (x/t)
(106)
( 3 ,o ,
): u
= u'
ρ = ρ ' , c o n s ta n t
= u2
ρ = ρ 2 , c o n s ta n t
We have f o r th e v e lo c it y o f p ro p a g a tio n o f th e d is c o n t in u it y :
(107)
The c o n d itio n o f c o n t in u it y a lo n g th e lin e s
( o ,l)
and (o ,2 ) demands:
(108)
(109)
From these e q u a tio n s we o b ta in :
-4 4 -
(110)
(111)
S u b tr a c tin g we o b ta in :
(112)
The r i g h t member o f th e above e q u a tio n in c re a s e s w ith in c re a s e
of ρ' i f ρ
'
andρ
1
ρ
2
must l i e
is g re a te r th a n ρ2 as is n e c e s s a ry .
in v a lu e ,
S ince ρ ' l i e s between
hence u 1 - u 2, g iv e n by th e above e q u a tio n ,
between i t s
v a lu e s f o r ρ'=ρ1 and ρ'=ρ2. T h e re fo re we have:
(113)
I f t h is in e q u a lit y is
ρ ' and when t h i s
fu lfille d
e q u a tio n (112) g iv e s a un iq ue v a lu e f o r
is known, e it h e r e q u a tio n s (110 o r (111) d e term in e
u ' and α 1 , α2, α3
a re th e n found fro m e q u a tio n s ( 1 0 7 ) ,
(108) and (109)
o f t h is pa ra g ra p h and a re in c o r r e c t o rd e r o f m agnitude w ith o u t f u r ­
th e r r e s t r i c t i o n s .
Riemann’ s f o u r t h case is n o t e s s e n t ia lly d i f f e r e n t from the
t h i r d and may b e o b ta in e d fro m i t by changing th e d ir e c t io n o f th e
x - a x is .
We may now sum up th e p re c e e d in g r e s u lt s .
For b r e v it y , l e t us
w r it e :
f ( ρ1) - f ( ρ2) = ± ∆ where th e s ig n o f ∆
th a t
is ta k e n such
∆ ( ρ 1 -ρ 2 )> o
Then we have:
Case I :
[u ] > R
Case I I :
- ∆ > [u ]
Case I I I :
R > [u ] > - ∆
, ρ1>ρ2
Case I v :
R > [u ] > - ∆
, ρ1> ρ2
W ith th e e x c e p tio n o f th e l i m i t a t i o n on u1 - u2 in Case I I
-4 5 -
to p re v e n t fo rm a tio n o f a vacuum, one o f these fo u r g iv e s an a c c e p ta b le
s o lu t io n f o r e ve ry s e t o f v a lu e s o f th e g iv e n q u a n t it ie s , u 1, u 2
, ρ1, ρ2.
The q u e s tio n s t i l l rem ains open as to w h e th e r t h is s o lu t io n is u n iq u e .
Riemann1 d id n o t d is c u s s t h is q u e s tio n
a
d
W
e
b
r2
a
lso
m
a
ke
sn
o
m
e
tio
n
f.H
a
d
m
r3
h
o
w
e
vr,te
a
sh
q
u
e
stio
n
q u ite th o ro u g h ly and shows th a t
s e v e ra l s o lu tio n s are sometimes p o s s ib le w h ic h s a t i s f y th e e q u a tio n s
o f m o tio n .
R iem ann's s o lu t io n a p p lie s s p e c i f i c a l l y to a d is c o n t in u it y
s e p a ra tin g two unbounded re g io n s in w hich u and ρ
a re c o n s ta n t, bu t
the s o lu t io n a ls o h o ld s f o r th e im m ediate n e ig hb orh oo d o f any d is c o n ­
t i n u i t y f o r the f i r s t in s t a n t o f tim e .
The g e n e ra l problem o f d is c o n ­
t i n u i t i e s w ith g iv e n i n i t i a l c o n d itio n s has n o t been s o lv e d .
2. Hadamard' s tre a tm e n t.
L e t us c o n s id e r the case o f d is c o n t in u it ie s as tre a te d by
Hadamard1
.
Ea ch o f th e waves o f r a r e f a c t io n tr e a te d by
Riemann
are c la s s if ie d two d is c o n t in u it ie s o f th e second o rd e r by Hadamard.
Hadamard t r e a t s o f th e r e s o lu t io n in to d is c o n t in u it ie s o f th e f i r s t
o rd e r.
Between th e two re g io n s in w h ich u and p and p have t h e i r
o r i g i n a l v a lu e s u1, ρ1, p 1and u 2 , ρ 2 , p 2 , we have an in te rm e d ia te
re g io n in w hich u, ρ and p have v a lu e s u ' , ρ ' , p ' . Hadamard c o n s id e rs
P o is s o n 's law to h o ld between the se th re e s ta t e s , namely pwy = c o n s ta n t
where w = 1 /ρ
is in tro d u c e d f o r co n ve n ie n ce .
He v is u a liz e s t h i s law
as a curve i n a p la n e in w h ich w and p are r e c ta n g u la r c o o rd in a te s . The
th re e p o in ts
(w 1 ,p1 ) ,
( w 2, p 2 ) , and (w'
,p' ) l i e
upon t h is c u rv e .
T his assum ption makes Hadamard's tre a tm e n t le s s g e n e ra l th a n Riem ann's
fo r i t
im p lie s t h a t o n ly th re e o f th e q u a n t it ie s , w1 , p 1 , w2 , p2 a re
a r b it r a r y .
In e f f e c t h e assumes t h a t t h is m o tio n is g e ne rated fro m a
gas i n i t i a l l y homogeneous such, i o r example, as by th e sudden move­
ment o f a p is to n ,
in w h ich case th e two s ta te s w ould H e on th e same
. Loc. c i t .
. Loc. c i t .
. Hadamard, Sur la P ro p a g a tio n des Undes, 1903 , p p .194-200
-4 6 -
a d ia ba t i c , n o t ta k in g acco un t o f H u g o n io t's o b je c tio n , We have fro m the
e q u a tio n o f Riemann (52) and (54) by e lim in a t io n o f 0
(114)
(115)
S u b tra c tin g we have:
(116)
where e ve ry c o m b in a tio n o f upper and lo w e r s ig n s b e fo re th e two
r a d ic a ls i s p e rm is s ib le b u t f o r e it h e r r a d ic a l th e s ig n is e it h e r
upper o r lo w e r f o r a l l th re e e q u a tio n s and t h i s r u le w i l l be extended
to the subsequent e q u a tio n s .
By t r a n s p o s it io n and s q u a rin g o f b o th
members we o b ta in f o r the above:
(117)
(118)
S q ua ring b o th o f th e se e q u a tio n s , we amalgamate th e two branches ex­
pressed by the ambiguous s ig n in t o a s in g le c u rv e , as fo llo w s :
(119)
(120)
These e q u a tio n s are e q u iv a le n t.
The p re c e e d in g e q u a tio n s , however, are
d i f f e r e n t as th e y d iv id e th e cu rve in to branches a t d i f f e r e n t p o in ts .
P 1 and P2 are q u a d ra tic e x p re s s io n s b u t P - P _ is li n e a r .
Thus t he
curve is o f th e second degree and hence, a coni c s e c tio n .
For P1 - P2+a2=o ,
from e q u a tio n (119) we' must have P1 = (p' - p 1
) (w1- w') = o and hence e it h e r
p' = p 1
, o r w' = w1. Since th e e x p re s s io n P 1- P2 + a2 occurs tw ic e as a
f a c t o r in equ a tio n (119) th e coni c is ta n g e n t to th e lin e s p ' = p 1and
w, = w ( , p a r a l l e l to th e c o o rd in a te axes a t t h e i r p o in ts o f in t e r s e c t io n
w ith the s t r a ig h t l i n e P, - P^+ a = o.
a c c o rd in g to e q u a tio n (120)
lik e w is e th e l i n e P, - Pa - a = o
in te r s e c t s th e c o n ic a t i t s
p o in ts o f
-4 7 -
tangency w ith th e lin e s p' = p2 and w' = w2
. S ince th e e q u a tio n o f th e curve
in v o lv e s the s in g le p a ra m e te r a , the cu rve is one o f a fa m ily o f c o n ic s
ta n g e n t to th e s id e s o f a re c ta n g le composed o f th e lin e s p ' = p1, w' = w1,
p' = p 2
, w' = w2 drawn w ith th e g iv e n p o in ts
o p p o s ite c o rn e rs .
(w1 ,p1, ) and (w2 , p2 ) a t
We e a s ily see th a t f o r a
com prised between o and
(p1 - p 2
) (w2 - w1 ) th e c o n ic is an e llip s e and i f a
value i t
surpasses t h is
is an h y p e rb o la . In th e accom panying Fi g . 4 , A 1and A2 are the
g iv e n p o in ts
(w1p 1 ) and (w2 p2) r e s p e c t iv e ly and A1B1A2B2 the re c ta n g le
to th e s id e s o f w hich the c o n ic is ta n g e n t.
as an e llip s e
in s c r ib e d in t h i s r e c ta n g le .
The c o n ic is re p re s e n te d
The cho rd C1D1has f o r i t s
e q u a tio n P 1 - P2 + a2 = o and th e chord C D has f o r i t s
e q u a tio n P1 - P2- a2 =o
such t h a t P1 - P2 = o re p re s e n ts the d ia g o n a l B1B2
T his c o n ic c u ts the a d ia b a tic between A1and A 2 in two p o in ts A' and A' ' .
Thus we have two va lu e s f o r th e i n te rm e d ia te s t a t e .
There is however a
f u r t h e r c r i t e r i o n to d is t in g u is h between the se two v a lu e s . C o nside r
the e q u a tio n s o f th e cu rve d iv id e d in t o b ra n ch e s.
The p o in t A ' l i e s
on
the a rc C1, D1, and th e p o in t A ' ' on th e a rc C2D2. F o r th e fo rm e r a r c ,
P1 - P2 + a 2
< o and f o r th e l a t t e r , P1 - P2 - a 2>o
We see by com paring the
s ig n b e fo re the r a d ic a ls in e q u a tio n s (117) and (118) w ith th e s ig n
b e fo re the same r a d ic a ls in e q u a tio n s (115) and (116) t h a t f o r th e a rc
C1D1, a (u '- u1
o
)>
and a (u'
- u 2 )> o and f o r th e a rc C2 D2 , a ( u ' -u 1) <o
and a (u' - u 2
)<o w h ile f o r th e r e s t o f the c o n ic , a ( u ' - u1)<o
a (u ' - u 2
) > o. L e t θ
1
andθ
2
and
be th e v e lo c it ie s o f p ro p a g a tio n o f the two
-4 8 -
d is c o n t in u it ie s w ith re s p e c t to an i n i t i a l s ta te in w h ich the d e n s ity
is
. Then th e e q u a tio n s o f c o n t in u it y
(52) become:
(121)
(122)
D iv id in g one e q u a tio n by th e o th e r we o b ta in :
(123)
F or b o th the a rc s C1D1and C2D2 u' - u 1 , and u' - u2 have the same s ig n
and hence we have:
(124)
A ls o , i f
th e c o n ic is an e l l i p s e , w'
l i e s between th e v a lu e s w1and w2
and hence w1 - w' and w2 - w ' have o p p o s ite s ig n s o r
T h e re fo re from th e p re c e e d in g e q u a tio n we have:
(125)
Thus on b o th o f the se a rc s , th e d i s c o n t in u it ie s t r a v e l in o p p o s ite
d ir e c t io n s , F or th e re m a in in g two a rc s o f ou r e l l i p s e , u ' - u 1and u ' - u 2
have o p p o s ite s ig n s and hence we have:
(126)
Thus on these a r c s, th e two d is c o n t in u it ie s t r a v e l in the same d ir e c t io n .
For ou r p o in ts A' and A'' on the a rc s C1D1 and C2D2 r e s p e c t iv e ly i t
c e s sa ry to in q u ir e i f
the c o n d itio n
θ1 < o is s a t is f ie d f o r b o th the se
p o in ts and i f f o r o n ly one, w h ich one.
th a t i f θ1 is to be n e g a tiv e w1<w' o r
is ne­
From e q u a tio n (1 2 1 ), we see
w1>w' a c c o rd in g as u' - u 1= +√
)
1
(P
or
u ' - u1
=- √ ( P 1 ) . F o r w 1 < w 2 , the fo rm e r a lt e r n a t iv e must be chosen and
t h is e n fo rc e s th a t th e in t e r s e c t io n w h ich is a c c e p ta b le must be on the
a rc C1D1 i f
the i n i t i a l d is c o n t in u it y is one o f co n d e n s a tio n (See
-4 9 -
e q u a tio n 98)
(a >o)
and on the a rc C2D2 i f
is one o f r a r e f a c t io n ( a< o)
the i n i t i a l d is c o n t in u it y
(See e q u a tio n s (1 0 4 )and(105).
Fo r w1> w2
th e l a t t e r a lt e r n a t iv e must be chosen and t h i s e n fo rc e s th e c h o ic e o f
th e a rc C2D2 i f the i n i t i a l d is c o n t in u it y is one o f co n d e n s a tio n (a> o)
and the a rc C1D1i f th e i n i t i a l d is c o n t in u it y is
We may p u t t h is in a l i t t l e
u ity
d i f f e r e n t fo rm .
one o f r a r e f a c t io n ( a < o ) .
I f the i n i t i a l d is c o n t in ­
is one o f c o n d e n s a tio n the p o in t A must be chosen n e a re r t h a t one
o f the two p o in ts A 1and A 2 w h ich corre spo nd s to the g re a te r d e n s ity ; i f
the i n i t i a l d is c o n t in u it y is one o f r a r e f a c t io n the p o in t A must be
n e a re r th e p o in t c o rre s p o n d in g to th e le s s e r d e n s ity .
Thus i f
the
c o n ic is an e llip s e and the o n ly in te r s e c t io n s o ccu r on the a rc s C1 D1
and C2D2 the s o lu t io n is u n iq u e .
i f th e c o n ic is an h y p e rb o la , w ' is
e x t e r io r to the range from w1to w2.
i n i t i a l d is c o n t in u it y is
w1 and w2 and i f
to b o th w 1 and
the
We know, how ever, t h a t i f the
one of c o n d e n s a tio n , w' is
i s c o n t i n u i t y is
d
i n f e r i o r to b o th
one o f r a r e f a c t io n , w' is s u p e r io r
w2 . S ince one in t e r s e c t io n w i l l always o ccu r on the
s id e i n f e r i o r to t h is range and one on the s id e s u p e r io r to t h i s ra n g e ,
t h is c r i t e r i o n serve s to d is t in g u is h th e c o r r e c t s o lu t io n .
more than two in t e r s e c t io n s of th e a d ia b a tic w it h th e c o n ic .
e may have
W
In the
case o f th e e llip s e p ic tu r e d in th e f ig u r e , we may have an in t e r s e c t io n
a lo n g the a rc C1D2 . That such p o in ts o f in t e r s e c t io n occu r may be seen
i f the e llip s e a p pro xim a te s u f f i c i e n t l y c lo s e ly i t s
l i m i t i n g p o s it io n
as a double li n e a lo n g the d ia g o n a l A1A 2
. These two in t e r s e c t io n s , as
we have seen by e q u a tio n (126) r e p re sen t m o tio n s f o r w h ich θ1 and θ2 have
the same s ig n .
I n the se cases the s o lu t io n is n o t u n iq u e b u t th e re are
s e v e ra l s o lu t io n s , a l l o f w hich a re p o s s ib le .
employ the c r i t e r i o n o f e n tro p y ,
Hadamard does n o t
f ######################
as
the a p p lic a t io n o f t h is p r i n c i p le had n o t been advanced a t the tim e he
2.
3•
U
5.
-5 0 -
w ro te h is book.
O th e rw is e , o f c o u rs e , he w ould r e je c t d is c o n t in u it ie s
o f r a r e f a c t io n as Zemplen1 , Jou q u e t2 and Duhem3 h a te done
(See page 2 5 ).
h . O b je c tio n to H u g o n io t's Law
A lth o u g h th e p r i n c i p le o f e n tro p y has been e x te n s iv e ly employ­
ed in c o n n e c tio n w ith f i r s t o rd e r d is c o n t in u it ie s ,
it
is d i f f i c u l t to
see how in the absence o f any d is s ip a t iv e fo r c e s , the gas in p a s s in g
th ro u g h a d is c o n t in u it y can s u f f e r a g a in in e n tro p y ,
d e c la re s t h a t i t c a n n o t.
lo r d R a y le ig h ^
He shawa th a t we may o b ta in the e q u a tio n o f
H ugoniot by c o n s id e rin g a wave o f perm anent wave form in w h ich the
t r a n s it io n between two s ta te s need n o t be a b ru p t b u t t h a t d is s ip a t iv e
fo rc e s must be a llo w e d to m a in ta in th e permanency o f the wave.
A
r e l a t io n id e n t ic a l to H u g o n io t’ s was found f i f t e e n y e a rs e a r l i e r by
Rankine
f o r a wave o f permanent w$ve form m a in ta in e d by h e a t con­
d u c tio n .
R a y le ig h says t h a t the e o u a tio n o f ehergy is s a t i s f i e d b u t
b u t n o t the second law o f therm odynam ics.
s id e o f the d is c o n t in u it y l i e ,
He s a y s ,in e f f e c t ,
The two s ta te s on e it h e r
in g e n e ra l, upon two d i f f e r e n t a d ia b a tic s .
th a t one cannot g e t from one a d ia b a tic to a n o th e r
w ith o u t a t r a n s f e r o f h e a t,
in th e d e r iv a t io n o f H u g o n io t’ s la w th e
in t e r n a l energy is e v a lu a te d fro m th e e x p re s s io n :
The d iffe r e n c e o f in t e r n a l e n e rg ie s in th e t r a n s f e r acro ss th e d is ­
c o n t in u it y is ta ke n as:
R a y le ig h p o in ts o u t t h a t t h is
in v o lv e d in th e passage a t
in v o lv e s the assum ption t h a t n o th in g is
w = c& from one a d ia b a tic to a n o th e r and th a t
Loc. c l 't .
HOC. e x t.
R a y le ig h , P roceedings o f th e R oyal S o c ie ty , 1910, yp 267-269.
R ankine, ” 0n the Thermodynamic Theory o f Waves o f F in i t e lo n g it u d in a l
D is tu rb a n c e " - P h il. T rans. 1870, v o l. 160. H a rt I I , p , 277.
- 51-
th e re is a c t u a lly re q u ire d the com m unication o f an i n f i n i t e s i m a l q u a n tity
o f h e a t. To g e t from one a d ia b a tic to th e o th e r a t r a n s f e r o f h e a t is
r e q u ir e d .
Heat can o n ly be gained a t the expense o f w ork and hence d is ­
s ip a t iv e fo rc e s must e n te r and an a b ru p t d is c o n t in u it y becomes th e n an
im p o s s ib ilit y ,
That d is s ip a t iv e fo rc e s c o u ld n o t be a llo w e d in a f i r s t
o rd e r d is c o n t in u it y is a ls o re c o g n iz e d by Duheni*' and f o r t h is
reason he
is a t a lo s s f o r an e x p la n a tio n o f the e n tro p y change f o r t h is case b u t
in s is t s t h a t , a c c o rd in g to th e c o n c lu s io n s o f Jouguet'-', h is r e s u lt s are
a p p lic a b le to a wave w ith v e ry sudden t r a n s i t i o n la y e r m a in ta in e d by
d is s ip a t iv e fo rc e s .
4.
i t h in k t h i s
s t i l l rem ains a d is p u te d p o in t .
Summary
I t has been shown t h a t d is c o n tin u o u s waves alw ays produce
v a r ia t io n s in the tim e d e r iv a tiv e s o f x , y , and z.
The d e r iv a tiv e s o f
one
the d e n s ity , o f o rd e r^ le s s th a n th e o rd e r o f d is c o n t in u it y are the f i r s t
to become d is c o n tin u o u s .T h e f i r s t o rd e r d is c o n t in u it y i s th e o n ly case
where th e re i s a d is c o n t in u it y in the d e n s ity i t s e l f .
I n e v e ry d i s ­
c o n tin u o u s wave th e c h a r a c t e r is t ic segment i s norm al to th e s u rfa c e
and in waves o f th e second o r h ig h e r o rd e r th e v e lo c it y o f p ro p a g a tio n
is t h a t o f o r d in a r y sound o f i n f i n i t e s i m a l a m p litu d e w ith re s p e c t to
the medium in i t s
c o n t in u it y .
a c tu a l s ta te on e it h e r s id e o f th e s u rfa c e o f d is ­
The c o n t in u it y and d yn am ical e q u a tio n s f o r a f i r s t o rd e r
d is c o n t in u it y were de ve lo p e d .
P’o r the a d ia b a tic law o f P o isso n the
v e lo c it y o f p ro p a g a tio n may be made as g re a t as we p le a s e by in c re a s in g
the p re s s u re on one s id e o f th e d i s c o n t i n u it y .
The energy e q u a tio n is
n o t s a t is f ie d by P oissons law o f p re s s u re and H u g o n io t's lav/ was de1. Loc. c i t .
2. Hoc. c i t .
-5 2 -
velop ed so as to s a t is f y t h is e q u a tio n .
The e n tro p y o f th e gas,
however, under t h is law s u f f e r s a change on p a s s in g th e d is c o n t in u it y .
The c o n d itio n o f in c re a s e o f e n tro p y a p p lie d to a c tu a l gases im p lie s
th a t o n ly a d is c o n t in u it y o f c o n d e n s a tio n is p o s s ib le .
The v e lo c it y o f
p ro p a g a tio n under n u g o n io t’ s law is alw ays g re a te r th a n t h a t o f o rd in a r y
sound o f in f i n i t e s i m a l a m p litu d e when b o th are measured w ith re s p e c t to
the medium in f r o n t o f th e wave and le s s th a n th e v e l o c i t y o f o rd in a ry
sound o f in f i n i t e s i m a l a m p litu d e when b o th are measured w ith re s p e c t to
the re g io n b e h in d th e wave.
The r e s o lu t io n o f an i n i t i a l d is c o n t in u it y
was d is c u s s e d a c c o rd in g to the method o f Riemann and o f Hadamard.
U sing
P o is s o n 's law i t was shown th a t a r e s o lu t io n in to two f i r s t o rd e r d is ­
c o n t in u it ie s s a t is f y in g the e q u a tio n o f c o n t in u it y and the dynam ical equa­
t io n c o u ld sometimes be made in more th a n one way.
The o b je c tio n o f
lo r d R a y le ig h to th e a p p lic a t io n o f n u g o n io t's law to a d is c o n t in u it y
was d is c u s s e d .
T his o b je c tio n , is i n su b sta n ce , t h a t the e n tro p y can
n o t in c re a s e w ith o u t th e e n tra n c e o f d is s ip a t iv e fo rc e s in w hich case
the t r a n s it io n between two s ta te s cannot be a b ru p t.
-5 3 C. I n f in it e s im a l A m p litu d e Sound Waves?
B e fo re ta k in g up th e s u b je c t o f sound waves o f f i n i t e
a m p litu d e i t
is p ro p e r to in d ic a t e th e g e n e ra l r e s u lt s a tta in e d
f o r in f i n i t e s im a l a m p litu d e waves as these waves a re r e a l l y the
f i r s t a p p ro x im a tio n to th e case o f f i n i t e
For convenience i t
a m p litu d e waves.
is custom ary to change from th e v a r ia b le p
to the v a r ia b le S d e fin e d by
(127)
The q u a n tity S is c a lle d the c o n d e n s a tio n .
The case where
u , v , w , s , and t h e i r f i r s t d e r iv a tiv e s w ith re s p e c t to x ,y ,z ,a n d
t are reg ard ed as so s m a ll t h a t th e second degree term s in the se
q u a n t it ie s may be dropped fro m th e e q u a tio n s o f m o tio n (1)
and ( 2 ) , may be c a lle d th e case o f sound waves o f in f i n i t e s i m a l
a m p litu d e .
The e q u a tio n s o f m o tio n the n become li n e a r .
I f the v e lo c it y has a p o t e n t ia l f u n c t io n , t h a t is
, if
th e re
e x is ts a fu n c tio n (p such t h a t : (128)
th e m o tio n is found to be i r r o t a t i o n a l .
When the se v a lu e s
o f u ,v ,w , a re s u b s titu te d in th e dyn am ical e q u a tio n s ( 2 ) , we
o b ta in by in t e g r a t io n and d is c a r d in g o f second degree term s
f o r the case o f in f i n i t e s i m a l a m p litu d e waves th e s in g le e q u a tio n :
(129)
where
C =
is to be c o n s id e re d c o n s ta n t s in c e th e d e n s ity
changes are s m a ll.
o f c o n t in u it y
W ith th e same s u b s t it u t io n
the e q u a tio n
(1 ) becom es:(130)
1.
The m a te r ia l f o r t h is s e c tio n is s e le c te d from
Lamb's H ydrodynam ics,1906, pp. 4 5 4 ,4 5 5 ,4 6 6 -4 7 2 .
-5 4 The e lim in a t io n o f S between e q u a tio n s (129) and (130) g iv e s :(131)
I f the gas is c o n s id e re d as unbounded and th e i n i t i a l
v a lu e s o f the v e lo c it y and d e n s ity a re g iv e n , th e i n i t i a l
v a lu e o f ϕ is every where g iv e n by in t e g r a t io n o f any one o f
e q u a tio n s (1 2 8 ), th e c o n s ta n t o f in t e g r a t io n re m a in in g a r b i t r a r y ,
and the i n i t i a l v a lu e o f ∂ ϕ / ∂ t is everyw here g iv e n by v ir t u e
o f e q u a tio n (1 2 9 ).
We may w r it e f o r the se i n i t i a l c o n d it io n s : (132)
(133)
The v a lu e o f ϕ a t any p o in t P a t a tim e t,
the o r ig in a l v a lu e s o f ϕ
about the p o in t P.
ifϕ
and ∂ϕ
/t
is a f u n c tio n o f
on a sphere o f ra d iu s c t
The e f f e c t a t P , m oreover, is
unchanged
and ∂ϕ/∂t are c o n s ta n t and eq u a l to t h e i r re s p e c tiv e mean
v a lu e s on th e sphere.
L e t ϕm and (
/t)m d e s ig n a te th e mean
∂ϕ
v a lu e s o f ϕ and ∂ ϕ /∂ t r e s p e c t iv e ly , on a sphere o f ra d iu s r
about th e p o in t P in the o r i g i n a l c o n d itio n .
I f co d e s ig n a te
a s o lid a n g le , we h a v e :(134)
(135)
where P is th e p o in t x , y , z , where r is the ra d iu s v e c to r from
t h is p o in t and
1, m, n i t s
d ir e c t io n c o s in e s .
The e f f e c t a t P
is the same as i f we were g iv e n a s y m m e tric a l s p h e r ic a l m o tio n
w ith i n i t i a l v a lu e s o f th e v e lo c it y p o t e n t ia l and i t s
d e r iv a t iv e , ^ =
v) and | ^ * l = / I (.>■) .
tim e
The v a lu e o f <f) as a fu n c tio n
o f t a t th e p o in t P in a s y m m e tric a l s p h e r ic a l m o tio n about t h is
p o in t w ith th e above i n i t i a l c o n d itio n s i s : (136)
-5 5 -
S u b s t it u t in g th e v a lu e s o f Γ and Λ
from e q u a tio n s (134)
and (135)
in e q u a tio n (136) and s in c e :r = c t , 1 = s i n θ cosϕ , m = s i n θ s i nϕ ,n = cosθ ,dω= s in θd θ d ϕ
we o b t a in : (137)
T h is is a com plete s o lu t io n f o r an i r r o t a t i o n a l sound wave o f
in f i n i t e s im a l a m p litu d e in an unbounded gas.
The s o lu t io n f o r
a plan e wave is
(138)
The s o lu t io n f o r a s y m m e tric a l s p h e r ic a l wave is :(139)
In b o th cases th e fu n c tio n s f and F a re to be d e te rm in e d by
the i n i t i a l c o n d itio n s .
-5 6 -
D. C ontinuous F in i t e A m p litu d e Sound Waves.
I s h a ll now d is c u s s c o n tin u o u s f i n i t e
a m p litu d e sound waves,
in t h is case no s o lu t io n has been o b ta in e d f o r th e e q u a tio n s in t h e i r
g e n e ra l form (1) and (2) o r (12) and ( 1 3 ).
I s h a ll , th e r e fo r e , c o n s id e r
f i r s t the case o f p la n e waves o r r e c t i l i n e a r d is tu rb a n c e s o f f i n i t e
a m p litu d e .
I.
Plane Waves
For p la n e waves we must o b ta in a s o lu t io n o f e q u a tio n s (3) and
(4) o r (15)and (1 6 ).
I n e it h e r system o f r e p r e s e n ta tio n th e re are b u t
two independent v a r ia b le s x and t o r a and t .
W hichever o f the two
dependent v a r ia b le s we w ish to c o n s id e r may be c o n s id e re d as a space
c o o rd in a te w ith the in de pe nd en t v a r ia b le s
th e o th e r two c o o rd in a te s .
A s o lu t io n o f the d i f f e r e n t i a l e q u a tio n s co rre sp o n d s th e n g e o m e tr ic a lly
to a s u rfa c e in w h ich these a re the c o o rd in a te s .
I s h a ll now d is c u s s
the p r o p e r tie s o f th e s o lu tio n s o f the e q u a tio n s o f m o tio n and o b ta in
s o lu tio n s f o r d i f f e r e n t c o n d itio n s .
1. G eneral Theory o f C h a r a c te r is t ic s f o r P lane WavesiL e t us c o n s id e r e q u a tio n s (15) and (16) f o r pla n e waves.
In s te a d o f e x p re s s in g p as a fu n c tio n o f ^
as in e q u a tio n ( 2 2 ) , l e t
us adopt a new v a r ia b le .
and express p as a f u n c tio n o f t h is v a r ia b le , namely
(140)
E lim in a tin g j °
between e q u a tio n s (15) and (16) and s u b s t it u t in g the
above v a lu e o f p, we o b ta in :
P la c in g - ---- w fa *
=
(wj we o b ta in :
(141)
L. Hadamard, lo c . c i t .
pp. 154-159, 173-175
-5 7 T h is is th e e q u a tio n to be s o lv e d to o b ta in x as a fu n c tio n o f a and t .
Since the v e lo c it y is u = ∂ x / ∂ t
and w = ∂ x / ∂ a , we have:
(142)
(143)
E lim in a tin g ∂2x/∂a
2 and ∂2x/∂t2
between e q u a tio n s (141)
(142) and (143)
we o b ta in :
(144)
The e q u a tio n :
(145)
When the s u rfa c e in t e g r a l is known, d e te rm in e s two f a m ilie s o f curve s
on t h is s u rfa c e , one curve o f each fa m ily p a s s in g th ro u g h each p o in t
on th e s u rfa c e .
The curve s th u s d e fin e d are c a lle d th e c h a r a c t e r is t ic s
o f t h is s u rfa c e .
f in ite .
I f o u r s o lu t io n is to be c o n tin u o u s , ∂2x/∂a∂t
must be
A lo n g one o f th e c h a r a c t e r is t ic s d e fin e d by (1 4 5 ), th e deno­
m in a to r o f th e r i g h t member o f e q u a tio n (144) is
be f i n i t e ,
zero and i f ∂2x/∂a∂t is to
the n u m e r a to r o f th e r i g h t member o f e q u a tio n (144) must
a ls o be z e ro .
T h e re fo re we must have
(146)
S u b s t it u t in g th e v a lu e o f da from e q u a tio n (19) in t h is e q u a tio n we
o b ta in
(147)
I n t e g r a t in g we o b ta in :
(148)
T his is a d i f f e r e n t i a l e q u a tio n w hich must be s a t i s f i e d a lo n g th e
c h a r a c t e r is t ic s o f o u r s u rfa c e in t e g r a l.
l e t us c o n s id e r the c o n d itio n th a t a second s u rfa c e in t e g r a l
-5 8 -
may ex i s t ta n g e n t to the f i r s t a lo n g a c u rve Γ
. Now the e q u a tio n o f
th e ta n g e n t p la n e to th e s u rfa c e in t e g r a l a t the p o in t x , a
, t
, i s:
(149)
Now a lo n g the cu rve Γ ,
the ta n g e n t p la n e s f o r th e two s u rfa c e s c o in c id e
and from the above e q u a tio n we see t h a t w and u must be th e same f o r
th e two s u ffa c e s a lo n g t h is c u rv e ,
i s o f the f i r s t o rd e r,
a lo n g the curve
in d e te rm in a te .
7 )V
i f th e c o n ta c t o f th e two s u rfa c e s
w i l l be d i f f e r e n t f o r the two s u rfa c e s
and hence i t s
e x p re s s io n fro m e q u a tio n (18) must be
T h is is o n ly in d e te rm in a te a lo n g a c h a r a c t e r is t ic and hene<
the curve JT* must be a c h a r a c t e r is t ic .
between two s u rfa c e in t e g r a ls i t
must be a c h a r a c t e r is t ic .
F or any f i n i t e
o rd e r o f c o n ta c t
i s p ro ve n t h a t th e cu rve o f tangency
The c h a r a c t e r is t ic
is th u s th e lo c u s ctf a
d is c o n t in u it y o f th e second o r h ig h e r o rd e r and e q u a tio n (145) agrees
w ith e q u a tio n (47) and shows t h a t the d is c o n t in u it y is p»opagated w ith
the o r d in a r y v e lo c it y o f sound o f in f i n i t e s i m a l a m p litu d e w ith re s p e c t
to the gas.
j
2. S in g le P ro g re s s iv e Yaves
A s in g le p ro g re s s iv e wave may be d e fin e d as one w h ich is p r o ­
pagated in t o a re g io n a t r e s t .
A d o p tin g o u r g e o m e tric a l re p r e s e n ta tio n ,
the re g io n a t r e s t is re p re s e n te d by the p la n e x = a .
A s in g le p r o ­
g re s s iv e wave, th e n , is one re p re s e n te d by a s u rfa c e in t e g r a l ta n g e n t
to th e p la n e x= a .
a . E arnshaw 's C o n d itio n 1
L e t us c o n s id e r a s u rfa c e in t e g r a l ta n g e n t a lo n g a c h a ra c te r ­
is t ic
o f th e f i r s t fa m ily o f e q u a tio n ( 1 4 5 ) ,nam ely:
(150)
Since w = 1 a lo n g t h is c h a r a c t e r is t ic w h ich p e r ta in s to the re g io n a t
. Hadamard, lo c . c i t . pp 174. 195
R a y le ig h , Theory o f Sound Y o l . I I pp . 34, 35.
-5 9 -
r e s t in w hich
is tic
x= a .
From any p o in t o f th e s u rfa c e , draw a c h a ra c te r ­
o f th e fa m ily o p p o s ite to t h a t o f th e c h a r a c t e r is t ic o f ta n g e n cy.
T h is c h a r a c t e r is t ic w i l l c u t the c h a r a c t e r is t ic o f tangency in a
p o in t P.
A long th e fo rm e r c h a r a c t e r is t ic we must have (e q u a tio n 14 8):
u = -X (w) + G
and a t the p o in t P, s in c e u = o, w = 1, we have:
o = -X (1) + C
T his de te rm in e s th e c o n s ta n t C and th e c o n d itio n a lo n g th e fo rm e r
c h a r a c t e r is t ic becomes:
u = X ( l)
- X(w)
(151)
Since t h is c h a r a c t e r is t ic was a r b i t r a r i l y chosen and s in c e e ve ry p o in t
on the s u rfa c e must have a c h a r a c t e r is t ic o f t h is fa m ily p a s s in g th ro u g h
it,
th e above is a c o n d itio n to be f u l f i l l e d
the wave.
The c h a r a c t e r is t ic o f tan ge ncy
o ve r th e whole s u rfa c e o f
g iv e s th e m o tio n o f the
f r o n t o f th e wave and fro m (150) we see th e v e l o c i t y o f th e f r o n t o f
the wave is p o s it iv e o r t h a t th e wave is a p o s it iv e p ro g re s s iv e wave.
E q u a tio n (151) is c a lle d E arnshaw 's c o n d itio n f o r a p o s it iv e p ro g re s s iv e
wave.
I f the c h a r a c t e r is t ic o f tangency be lo ng s to the o th e r fa m ily
o f e q u a tio n (145) t h a t i s ,
corre spo nd s to
we may s i m i l a r l y show t h a t t h is
s u rfa c e in t e g r a l s a t i s f i e s
u = X (w) - X (1)
the r e l a t io n :
(152)
The p ro p a g a tio n in the n e g a tiv e d ir e c t io n is n o t e s s e n t ia lly d i f f e r e n t
from t h a t i n the p o s it iv e d ir e c t io n s in c e th e one case becomes the
o th e r by r e v e r s in g the d ir e c t io n s o f th e a - and x - axes.
T h is r e v e r s a l
o f axes im p lie s a s u b s t it u t io n o f a = -a , x = -x , and u = -u and we see from
the above fo rm u la e t h a t th e one case tra n s fo rm s in to th e o th e r .
Thus the
two cases d i f f e r o n ly as an image in a p la n e m ir r o r d i f f e r s fro m i t s
-6 0 -
o b je c t.
Ho?/ from e q u a tio n (1 4 8 ), X ’ (w) - t / V (w) and from th e rem ark
p re c e e d in g e q u a tio n (1 4 1 ), [jj (w) = -
(w) . I f th e p re s s u re in / 0
creases w ith in c re a s e o f d e n s ity we see from e q u a tio n s (15) and (140)
th a t j) (w) is n e g a tiv e and hence
r e a l and p o s it iv e .
^
(w) is p o s it iv e and hence X ’ (w) is
X(w) is th e n an in c re a s in g f u n c tio n o f w.
Prom
e q u a tio n s (151) and (152) f o r p o s it iv e and n e g a tiv e waves r e s p e c tiv e ly
we see th a t f o r th e fo rm e r u decreases w it h w and f o r th e l a t t e r u i n ­
creases w ith w o r from e q u a tio n (15) we see t h a t f o r th e fo rm e r u i n ­
creases w ith
f
and f o r th e l a t t e r u daereases w ith
n e c e s s a r ily fo llo w s i f
f 1 . T h is r e s u lt
the r e s is ta n c e o ffe re d by the gas to th e m o tio n
o f th e p is to n , used to g e n e ra te a s in g le p ro g re s s iv e wave i n a c y lin d ­
r i c a l tu b e , in c re a s e s w ith i t s
speed i f t h i s is measured p o s it iv e i n
a d ir e c t io n com pressing the gas.
p is to n is pS where S
The re s is ta n c e to th e m o tio n o f the
is the c ro s s s e c tio n are a o f th e p is to n .
I f th e
re s is ta n c e in c re a s e s w ith u , p must in c re a s e w ith u and s in c e p is an
in c r e a s in g f u n c tio n o f
f
, p must in c re a s e w ith u .
We may o b ta in E arnshaw 's c o n d itio n Bp a method due to Lo rd
R a yle ig h ? - 1 e may im agine an;y p o in t w h ich we are exa m in in g to be b ro u g h t
to r e s t f o r the in s t a n t by im p re s s in g upon th e gas as a whole a u n i­
form v e lo c it y .
in th e neig hb orh oo d o f t h is p o in t th e c o n d itio n is th a t
o f a wave o f i n f i n i t e s i m a l a m p litu d e and the e q u a tio n s g iv e n in
s e c tio n C o b ta in ,
f o r a p o s it iv e p ro g re s s iv e i n f i n i t e s i m a l wave E=o in
e q u a tio n (138) and s u b s t it u t in g the v a lu e o f
(j) from t h i s e q u a tio n
f o r u , (128) and in the e q u a tio n i o r S, (1 2 9 ), we o b ta in on e lim in a t in g
f'
between th e r e s u lt in g e q u a tio n s :
1 .R a y le ig h , Theory o f Sound, 1896, V o l. I I , p p . 24 -35 .
-6 1 -
o r s in c e
c= √(ϕ'(ρ)), u/s = √(ϕ'(ρ))
Now in the wave o f f i n i t e
(1 5 3 )
a m p litu d e we must c o n s id e r i n f i n i t e s i m a l
v e lo c it ie s and co n d e n sa tio n s r e l a t iv e
to th e p o in t b ro u g h t to r e s t .
We m ust, t h e r e fo r e , use du f o r u and, by e q u a tio n (127), dρ/ρ f o r
and we must have f o r th e v i c i n i t y o f e ve ry
S
p o in t in th e wave:
(154)
I n t e g r a t in g t h i s e x p re s s io n from a p o in t on th e c h a r a c t e r is t ic o f
tangency where u = o and ρ
= ρ0 to any p o in t in the wave h a vin g a
v e lo c it y u and a d e n s ity ρ we o b ta in :
(155)
w hich is e q u iv a le n t to e q u a tio n (151) and i s Earnshaw’ s c o n d itio n f o r
a p o s it iv e p ro g re s s iv e wave.
b . G eneral S o l u t i o n and P r o p e r tie s o f a S u rfa ce I n t e g r a l.
S ince a p o s it iv e and
n e g a tiv e wave are n o t e s s e n t ia lly
d i f f e r e n t , l e t us c o n s id e r a p o s it iv e wave f o r w hich e q u a tio n (151)
h o ld s .
tic
A lo ng a c h a r a c t e r is t ic o f th e same fa m ily as th e c h a r a c te r is ­
o f ta n g e n cy,
s ig n .
(e q u a tio n 150) we have e q u a tio n (145) w ith th e upper
To t h is corre spo nd s e q u a tio n (148) w ith the upper s ig n , t h a t i s :
u = X ( w) + C
For th e s u rfa c e o f the wave in g e n e ra l we have e q u a tio n (1 5 1 ) nam ely:
u = X ( l)
- X(w)
A dding and s u b tr a c tin g , we see t h a t u and w a re b o th c o n s ta n t f or
p o in ts on the same c h a r a c t e r is t ic ; f o r w h ic h we have e q u a tio n (145) w ith
the upp e r s ig n .
T h is shows t h a t c o n s ta n t va lu e s o f u and w are tr a n s ­
m itte d th ro u g h the gas w ith
to th e i n i t i a l s t a t e .
v e l o c i t i e s e q u a l to √(ψ(w)) w ith re s p e c t
T his is th e v e lo c it y o f i n f i n i t e s i m a l sound in
a gas o f d e n s ity c o rre s p o n d in g to th e v a lu e o f w.
F or a p o s it iv e p ro g re s s iv e wave we have:
-6 2 -
dx=w∙d a + u ∙d t s in c e w = ∂ x /∂a
and u =
∂x/∂t
S u b s t it u t in g th e v a lu e o f u from e q u a tio n (151) we o b ta in :
dx = w∙ d a + [ X ( l)
- X (w )]
The s u rfa c e in t e g r a l is th e n found
x = w a + [X ( l )
o = a - X'
- X(w)]
(156)
by e lim in a t in g w between th e eq ua tion s
t +ϕ
)
(w
(157)
(w) t + ϕ ' (w)
When the a r b it r a r y f u n c tio n ϕ
c a r r ie d o u t.
dt
(158 )
is s p e c if ie d , th e e lim in a t io n may be
T h is is E a rn ’ shaw’ s1
method o f s o lu t io n .
E q u a tio n (157)
is th e e q u a tio n o f a ta n g e n t p la n e , as we see by com paring w ith e q u a tio n
(149) to th e s u rfa c e and, s in c e i t
the s u rfa c e is d e v e lo p a b le .
depends o n ly upon one p a ram e te r w,
I f u o r w is ta ke n as dependent v a r ia b le
in s te a d o f x , the dependent v a r ia b le is c o n s ta n t a lo n g th e ch a ra ct e r i s t i c s
o f th e fa m ily g iv e n by e q u a tio n (145) w ith the up pe r s ig n .
a c t e r is t ic s are the n h o r iz o n t a l g e n e ra tric e s o f a r u le d
The c h a r­
s u rfa c e i f
the a t - pla n e be ta k e n as h o r iz o n t a l.
E lim in a t in g a between e q u a tio n s (156) we o b ta in :
x - [X ( l )
- X (w) + w X ' ( w ) ] t =
S u b s t it u t in g X ( l)
ϕ (w) - W ϕ ' (w)
(1 57A)
- X(w) = u from (151) and wX ’(w) = √(ϕ'(ρ))
from e q u a tio n s '(1 4 8 ) an d(1 5 ’ )
we o b ta in :
(158A)
Where £
o f w.
is an a r b it r a r y f u n c tio n o f
a s in c e by (151) u in a fu n c tio n
R e ve rsin g th e f u n c tio n a l r e l a t i o n , we o b ta in :
(159)
T his i s known as P o is s o n 's in t e g r a l a lth o u g h P o isso n 3 o b ta in e d i t
f o r B o y le ’ s law (21) f o r w h ich ϕ '(ρ )= C2 is c o n s ta n t.
o n ly
T h is e q u a tio n
shows th a t va lu e s o f u are p ro p a g a te d w ith v e l o c i t i e s u + √(ϕ'(ρ)) w ith
re s p e c t to a fix e d s e t o f axe s.
From L o rd R a y le ig h ’ s2 method o f a n a ly s is
1. Earnshaw, P roceedings o f th e R oyal S o c ie ty , Jan 6, 1859.
2.R a y leig h , Theory o f Sound V o l I I 1896 pp 33 ,3 4
3. P o is s o n , Memoire s u r l a T h e o rie du Son, J o u rn a l de l'E c o le P o ly te c h n iq u e 1
9
3
p
l.7
o
,V
0
8
-6 3 -
we see t h a t t h is must be so s in c e any p o in t may be b ro u g h t to r e s t by im­
p re s s in g a c e r t a in v e lo c it y upon th e gas as a whole and in th e n e ig h ­
borhood o f t h is p o in t th e m o tio n must be t h a t o f a wave o f i n f i n i t e s im a l
a m p litu d e and henee v a lu e s o f th e v e lo c it y a re pro p a g a te d w ith a v e lo c
i t y
, t h a t o f o r d in a r y sound o f i n f i n i t e s i m a l a m p litu d e .
v e lo c it y o f p ro p a g a tio n in the f i n i t e
Hence the
a m p litu d e wave must be th e
v e lo c it y o f th e gas u p lu s the v e lo c it y o f o rd in a r y sound in the gas
w hich has th e v e lo c it y u .
Weber
ta ke s e q u a tio n ( 158A) and changes to a new v a r ia b le
(160)
from e q u a tio n (154)
. Then e q u a tio n (158A) becomes:
(161)
T his shows th a t V|
is c o n s ta n t a lo n g a s t r a ig h t l i n e i n the x t - p la n e ,
the c o ta n g e n t o f whose a n g le w ith the x - a x is i s - 7^ •
c . S o lu tio n w ith G iven M o tio n o f P is to n .
£
Hadamard t r e a t s the pro blem o f the m o tio n o f a gas i n i t i a l l y
a t r e s t in a c y l i n d r i c a l tube when th e m o tio n is gene rated by th e m o tio n
o f a p is to n .
I f th e re a re two m oving p is to n s e n c lo s in g th e gas h is
s o lu t io n h o ld s o n ly f o r such re g io n s and such tim e s as th e wave rem ains
a s in g le p ro g re s s iv e one, th e e n c o u n te r o f the waves i s s u i n g from the
two p is to n s p ro d u c in g a new p ro b le m .
..h e re ve r a r e f le c t e d wave e n te rs
h is s o lu t io n lik e w is e does n o t a n p ly .
L e t th e i n i t i a l a b s c is s a o f the
p a r t ic le s a t th e s u rfa c e o f th e p is to n c o n s id e re d be a=o.
L e t the
s u b s c r ip ts zero denote q u a n t it ie s p e r t a in in g to th e m o tio n o f the
p is to n o r the m o tio n o f th e gas a t a=o.
be g iv e n by:
L. Weber, lo c . c i t . p p .475-477
5. Hadamard, lo c . c i t . pp. 177-178
L e t th e m o tio n o f th e p is to n
-6 4 x o = F (to )
(162)
Then the v e lo c it y is g iv e n by
u o = F'
(t o )
(1 6 3 )
where F ' is th e d e r iv a t iv e fu n c t io n o f F .
Prom the f i r s t p a ra g ra p h o f
s e c tio n b above we have seen t h a t v a lu e s o f u o and wo a re p ro p a g a te d w ith
v e lo c it ie s √
)
o
w
(ψ
w it h re s p e c t to th e gas in i t s
i n i t i a l s ta te .
A c c o rd in g ly , a t a tim e t > t o th e v e lo c it y uo and c o rre s p o n d in g v a lu e o f
w o has been communicated to a p a r t i c l e whose i n i t i a l a b s c is c a is :
(164)
The a c tu a l v e lo c it y o f p ro g a g a tio n , a c c o rd in g to e q u a tio n (159) and
the succe ed in g d is c u s s io n is u + √(ϕ'(ρo)) o r from th e sta te m e n t im m e d ia te ly
f o llo w in g e q u a tio n (157 ) i t
is u o + wo X ' (wo) o r from e q u a tio n (148)
u o + wo√(ψ(wo)) . Thus th e a c tu a l a b s c is c a c o rre s p o n d in g to a v a lu e o f u
and w
is :
(165)
Now w ois expressed in term s o f u oby e q u a tio n (151) and u o is expressed
i n term s o f t o by e q u a tio n (163)
. Thus e q u a tio n s (164) and ( 165 )
b e sid es x , a, and t in v o lv e o n ly t oand the e lim in a t io n o f
t o between
these e q u a tio n s g iv e s the sou gh t f o r i n t e g r a l .
d . S in g u la r it y i n the S o lu tio n
We may perhaps most e a s ily p ic t u r e the s i n g u la r i t y in v o lv e d
i f we adopt Webers1 system o f r e p r e s e n ta tio n , e q u a tio n (1 6 1 ).
I n i t i a l l y we are g iv e n
as a f u n c tio n o f x , t h a t i s ,
lo n g the x a x is in th e x t p la n e .
th e v a lu e s o f η a­
Prom each p o in t on th e x - a x is c o n s tru c t
a s t r a ig h t l i n e making an angle w ith t h i s ax is whose co ta n g e n t is
η
Then as lo n g as two s t r a ig h t lin e s do n o t in t e r s e c t in a p o in t on the
p o s it iv e s id e o f the x - a x is , the s o lu t io n is u n iq u e , t h a t is
1 . Weber, lo c . c i t . p . 477.
^
is
.
-6 5 -
u n iq u e ly de te rm in e d f o r a l l p o in ts f o r p o s it iv e va lu e s o f t .
In g e n e ra l,
however, t h is f a m ily o f s t r a ig h t lin e s w i l l in t e r s e c t on the p o s it iv e
s id e o f th e x - a x is .
These lin e s w i l l o n ly f a i l to in t e r s e c t in th e
re g io n o u ts id e o f th e envelope o f th e f a m ily o f s t r a ig h t lin e s and i t
is o n ly th e re th a t η
is u n iq u e ly d e te rm in e d .
by the e lim in a t io n o f η
x = G(η
)
T h is envelope is g iv e n
between the e q u a tio n s ,
- η G '(η )
t = - G’ ( η )
(166)
A n o th e r in t e r e s t in g method o f v is u a liz in g the p ro ce ss is th a t
o f Lord R a y le ig h .2
A t any in s t a n t p l o t a curve w ith th e v e lo c it y
as o rd in a te and the a c tu a l a b s c is c a
curve is deformed w ith tim e .
as a b s c is c a . L e t us see how t h is
A v a lu e u is pr o pagated w ith a v e lo c it y
u + √ (ϕ '(ρ ))(See e q u a tio n (159) and f o llo w in g d is c u s s io n ) .
wave w i l l o n ly r e t a in i t s
T h is v e lo c it y
o r i g i n a l form i f u + √(ϕ'(ρ)) i s con s t a n t.
S u b s t it u t in g the v a lu e o f u from e q u a tio n (1 5 5 ), th e c o n d itio n f o r a
permanent wave form becomes:
D i f f e r e n t ia t in g and m u lt ip ly in g by
we have:
I n t e g r a t in g we have: √ ( ϕ '(ρ )) = B /ρ
S quaring we have: ϕ ' ( ρ ) = B 2 / ρ 2
I n t e g r a t in g a g a in we have
p = ϕ (p )= A - (B2
)
/ρ
(167)
I t is o n ly under t h i s law o f p re s s u re t h a t a v e lo c it y wave w i l l r e t a in
it s
o r i g i n a l fo rm . T his law is n o t f u l f i l l e d
the v e lo c it y wave is deform ed.
f o r any a c tu a l gas and hence
Fo r B o y le ’ s law (20) and P o is s o n 's
l . R a y le ig h , P roceedings o f th e R oyal S o c ie ty , J u ly 8, 1910, p p . 2 4 8 ,2 4 9 ,2 5 2 .
-6 6 a d ia b a tic law (2 1 ), u
tim e t ,
is an in c re a s in g fu n c tio n o f u .
A fte r a
e ve ry p o in t on th e v e lo c it y curve w i l l have moved fo rw a rd
p a r a l le l to th e x - a x is a d is ta n c e
u +
t . A lo n g a p o r t io n o f th e
v e lo c it y curve w hich s lo p e s downward in th e p o s it iv e d ir e c t io n , the
v e lo c it y o f p ro p a g a tio n , u +
, is g re a te r th e h ig h e ru p a p o in t is
on th e v e lo c it y curve such t h a t the s lo p e o f the wave becomes e ve r more
steep u n t i l a t a c e r t a in tim e i t becomes v e r t i c a l a t a c e r t a in p o in t
a f t e r w hich a v e r t i c a l l i n e
c u ts th e cu rve in more tha n a s in g le p o in t
and th e v a lu e o f u is no lo n g e r u n iq u e a t a p o in t .
Hence the s o lu t io n
must be d is c a rd e d beyond the p o in t where th e curve a c q u ire s a v e r t i c a l
ta n g e n t.
The o n ly ty p e
o f v e lo c it y curve w hich may escape u ltim a te
d is c o n t in u it y is one w h ich has no downward s lo p e b u t has an upward s lo p e
from - ao to
+ « in th e p o s it iv e d ir e c t io n o f th e x - a x is .
curve is lim it e d
, however, as to th e d u r a tio n o f i t s p re v io u s e x is te n c e
and when p r o je c te d
c e r t a in tim e .
Such a
back in to n e g a tiv e tim e is d is c o n tin u o u s a t a
-6 7 3 . Waves, n o t S in g le P ro g re s s iv e .
a . G eneral S o lu tio n .
A more g e n e ra l s o lu t io n th a n t h a t f o r S in g le p ro g re s s iv e
waves was found by Riemann1 .
I t o c c u rre d to Riemann to adopt as de­
pendent v a r ia b le s in s te a d o f u andρ
in e q u a tio n s (3 ) and (4) th e
q u a n t it ie s r and s d e fin e d by:
2 r = f(ρ ) + u
(168)
2 s = f (ρ
(169)
) - u
where
f ( ρ ) = ∫√(φ'(ρ)) d lo g ρ
T his fu n c tio n
(170)
f ( ρ ) is e q u iv a le n t to th e f u n c tio n X (w) and we
see fro m e q u a tio n (148) t h a t r and s are c o n s ta n t a lo n g th e c h a ra c te r ­
i s t i c s o f th e su r f a ce i n t e g r a l. S o lv in g th e above equ a tio n s f o r f ( ρ )
and u we have:
f (ρ ) = r + s
(171)
u = r - s
(172)
D i f f e r e n t ia t in g these e x p re s s io n s , we have by means o f e q u a tio n (1 7 0 ):
(173)
The e q u a tio n s o f m o tio n (3) and (4) may be expressed in th e fo rm :
(174)
(175)
S u b s t it u t in g the e x p re s s io n s e v a lu a te d in e q u a tio n s (173) in e q u a tio n s
(174) and (175) we o b ta in :
Riemann, lo c . c i t .
-6 8 -
Adding and s u b tr a c tin g we o b ta in the e q u a tio n s :
(176)
(177)
Since u and ρ
a re g iv e n as fu n c tio n s o f r and S by e q u a tio n s ( 171)
and (172) the above e q u a tio n s a re the e q u a tio n s sought f o r in w hich r
and S re p la c e u and ρ as dependent v a r ia b le s .
L e t us form the t o t a l d i f f e r e n t i a l s o f r and S.
We have by
means o f e q u a tio n s (176) and (177)
(178)
(179)
To f in d th e v e lo c it ie s w ith w h ich r and S a re propagated we must f in d
the r e l a t io n b e tween dx and d t f o r d r = o and ds=o r e s p e c t iv e ly .
From e q u a tio n (178) and (179) we see t h a t
f o r d r= o ,
and f o r ds=o
dx/d
√
+
u
t=
(ϕ '(ρ ))
dx/d
t =
(180)
u -√(ϕ
'(ρ)
(181)
Thus v a lu e s o f r ar e pro p a g a te d in th e p o s it iv e d ir e c t io n w ith a ve­
l o c i t y u +√
)
'ρ
(ϕ
and v a lu e s o f 8 are propagated in th e n e g a tiv e d ir e c t io n
w ith a v e lo c it y -u+ √
).
'ρ
(ϕ
Suppose i n i t i a l l y we are g iv e n th e v e lo c it y
and d e n s ity as fu n c tio n s o f x .
Then by e q u a tio n s (168) and (169)
we are a ls o g iv e n r and S as fu n c tio n s o f X.
L e t us p l o t two c u rv e s ,
ta k in g x as absci s sa and r as o rd in a te i n the one ease and S i n th e
o th e r.
We are g iv e n the curve s f o r r and S a t t= o .
A f t e r a tim e d t , a
p o in t on the r - c u r v e has moved a d is ta n c e (u+ (√ϕ'(ρ))) d t p a r a l l e l to the
x - a x is and a p o in t on th e s -c d rv e a d is ta n c e (u -(√ϕ'(ρ)))d t p a r a l l e l to
the x - a x is .
By ta k in g s m a ll enough in cre m e n ts o f tim e and u s in g the
above c o n s tr u c tio n s u c c e s s iv e ly we may a p p ro xim a te as c lo s e ly as we
please the p ro g re s s o f the w a v e ..
-6 9 -
Under c e r t a in c o n d itio n s we may o b ta in a s o lu t io n w ith o u t
t h is te d io u s g ra p h ic m ethod.
The p r i n c i p le
o f t h is s o lu t io n o b ta in e d
by Riemann depends upona change to l i n e a r e q u a tio n s by in te rc h a n g e o f
the r o le s o f independent and dependent
v a r ia b le s .
T h is may alw ays be
accom plished i f the e q u a tio n s a r e o f the f o llo w in g ty p e , w hich is the
type to w hich e q u a tio n s (176) and (177) b e lo n g :
(182)
(183)
where the c o e f f ic ie n t s U(
---------- are fu n c tio n s o n ly o f u , and u 2
,
n o t o f x f and x ^ . We have,
S o lv in g f o r th e d i f f e r e n t i a l s dx
and d x fc we o b ta in :
E q u a tin g the c o e f f ic ie n t s o f du and a u z on b o th s id e s o f th e se e q u a tio n s
we o b ta in :
(184)
S u b s t it u t in g these v a lu e s in e q u a tio n s (182) and (1 8 3 ), th e f a c t o r
may be dropned and we o b ta in :
(185)
(186)
Independent and dependent v a r ia b le s have here been in te rc h a n g e d and
s in c e th e c o e f f ic ie n t s are fu n c tio n s o f th e indepen d e nt v a r ia b le s
o n ly th e e q u a tio n s a re li n e a r and have th e p ro p e rty t h a t the sum o f
any number o f s o lu tio n s is a ls o a s o lu t io n .
A p p ly in g t h is
tra n s fo rm a tio n
-7 0 -
to eq u a tio n s (176) and (177) we o b ta in :
(187)
(188)
These e q u a tio n s may be
p u t in the e q u iv a le n t form :
(189)
(190)
From e q u a tio n s ( 171) and (172) we have:
(191)
(192)
Subs t i t u t i n g these q u a n t it ie s i n th e r i g h t members o f e q u a tio n s ( 189)
and (190) we have:
(193)
(194)
S ince the r i g h t members o f these e q u a tio n s a re equ a l in m agnitude b u t
o p p o s ite in s ig n , the l e f t members are eq u a l i n m agnitude and o p p o s ite
in s ig n . Hence we have:
(195)
T his is th e ne ce ssa ry and s u f f i c i e n t c o n d itio n f o r the e x i s t ance o f a
f u n c tio n w d e fin e d by:
(196)
(197)
S u b tra c tin g we o b ta in :
(198)
S u b s t it u t in g the l e f t members o f e q u a tio n s (196) and (198) in e q u a tio n
(193) o r the l e f t members o f e q u a tio n s (197) and (198)
in e q u a tio n (194)
we o b ta in :
(199)
-7 1 -
We may w r it e f o r b r e v it y ,
(200)
where m by com parison w it h the above is a f u n c t io n o f ρ
and th e r e fo r e ,
by e q u a tio n (1 7 1 ), a f u n c tio n o f r + s. The f u n c tio n m depends on the
law o f p re s s u re assumed.
F o r B o y le ’ s la w ,
(20) we have:
(201)
and f o r Po is s o n 's a d ia b a tic la w ,
(21) we have:
(202)
Thus we have reduced o u r problem to th e s o lu t io n o f a second
o rd e r l i n e a r d i f f e r e n t i a l e q u a tio n (200) w it h th e c o n d itio n o f s a t is ­
f y in g a g iv e n i n i t i a l c o n d itio n . When w is found as a f u n c tio n o f r and
S as a s o lu t io n o f t h is
e q u a tio n , equ a tio n s (196) and (1 9 7 ), when r
and S have been e lim in a te d by means o f e q u a tio n s (168) and (1 6 9 ), g iv e
two e q u a tio n s between u , ρ , x
o b ta in u and ρ
and t and by s o lv in g s im u lta n e o u s ly we
as fu n c tio n s o f x
t
. As to th e i n i t i a l c o n d itio n to be
s a t is f ie d by e q u a tio n (2 0 0 ), we see from e q u a tio n s (196) and (197)
th a t i n i t i a l l y we must have:
(203)
(204)
Now u and ρ and hence, a c c o rd in g to e q u a tio n s (168) a n d (169) r and S are
in itia lly
g iv e n as fu n c tio n s o f X .
Hence, a c c o rd in g to
( 2 0 4 ), w
is determ ined by in t e g r a t io n , exce p t f o r an a d d it iv e a r b i t r a r y c o n s ta n t
w hich has no e f f e c t on the s o lu t io n , as a f u n c tio n o f x f o r t= o .
The e q u a tio n s :
r = c o n s ta n t
S = c o n s ta n t
d e fin e two f a m ilie s o f curve s in the x t - p l ane. Fo r a cu rve o f th e
f i r s t fa m ily we have f r om e q u a tio n s (178) and (179)
-7 2 -
(205)
and f o r a cu rve o f the second f a m ily :
(206)
L e t us c o n fin e o u r a t t e n t io n to a le n g th o f α β
∂r/∂x
and ∂s/∂x
do n o t change s ig n .
o f the x - a x is f o r w hich
Then a lo n g th e curve r= c o n s ta n t
S in c re a s e s o r decreases th ro u g h o
t and a lo n g th e curve S= c o n s ta n t, r
u
in c re a s e s o r decreases th ro u g h outa c c o rd in g to e q u a tio n s (£05) and (£06)
I f from e v e ry p o in t o f the lin e o ^ j3 w e draw a cu rve o f each f a m ily ,
curves form a n e tw o rk and f i l l a p o r t io n o f th e p la n e
these
w hich is
hounded by the x - a x is , and the cu rve s
/
/
r = r
S = S
/
,
/
where r
is th e va lu e o f r a t <A and S th e v a lu e o f S a t ^3 .
The curves r= c o n s ta n t and S = co nstan t may the n serve as c o o rd in a te in
t h is domain.
In a p la n e in w h ich r and S a re ta k e n as re c ta n g u la r
c o o rd in a te s and in w h ich th e cu rve s r= c o n s ta n t and S =constant are s t r a ig h t
lin e s p a r a l l e l to the c o o rd in a te axes, the l i n e
w hich has no maximum p o in t between th e p o in ts 0( and
becomes a curve C
(3
in s o lv in g e q u a tio n (200) we use Stokes theorem a p p lie d to
a p la n e in th e form :
(207)
where U and V are any c o n tin u o u s fu n c tio n s o f r and s and where the
Rouble in t e g r a l is ta k e n ove r a domain in th e rs p la n e around th e bound­
a ry o f w h ich the li n e
be l a t e r de term in ed
in t e g r a l is ta k e n . L e t v be a f u n c tio n o f r , s to
and l e t
(£08)
Prom the above by d i f f e r e n t i a t i o n we have:
(209)
-7 3 -
L e t us determ ine th e v a lu e o f v so as to s a t is f y th e e q u a tio n :
(210)
From t h is e q u a tio n and from e q u a tio n (200) , e q u a tio n (209) becomes:
(211)
S to k e 's theorem , e q u a tio n (207)
, th e n g iv e s :
(212)
in w h ic h thein t e g r a l is ta ke n around th e boundary o f a domain in the rs - p la n e
A p p ly in g t h is to th e domain α β ξ , bounded by the cu rve C and t h e s t r a ig h t
lin e s r = r ' , S = S'
we have:
(213)
By p a r t i a l in t e g r a t io n we have:
(214)
S u b s t it u t in g the va lu e o f t h is
in t e g r a l in
(213) we o b ta in :
(215)
L e t us s u b je c t v to th e f o llo w in g c o n d itio n s :
(216)
Then e q u a tio n (215) becomes:
(217)
Thus the problem is changed to one o f d e te rm in in g th e f u n c tio n v f o r
when v is known, Wξ, is de te rm in e d by the above fo rm u la from th e va lu e s
o f w,∂w
/r
and ∂ w / ∂ s on th e curve C.
L e t us now c o n s id e r th e d e te rm in a tio n o f th e f u n c t io n v .
T his f u n c tio n must s a t i s f y e q u a tio n (210) and boundary c o n d itio n s (2 1 6 ).
The boundary c o n d itio n s may be s im p lif ie d by a change o f dependent
v a r ia b le .
L e t us s u b s t it u t e
-7 4 -
v = eν V
(218)
where ν is a f u n c tio n o f r and S s t i l l to be d e te rm in e d .
(216)
s u b s t it u t io n the boundary c o n d itio n s /b e c o m e
S ince m is a f u n c t io n o f σ
= r + S, i f we ta k e ν
. W ith t h is
= -∫
dth e above
'm
σ
c o n d itio n s become:
V =1
fo r
σ = σ'
where σ' = r ' + S'
and these are o b v io u s ly e q u iv a le n t to th e c o n d itio n :
V=1 f o r r = r ' and S = S' , t h a t is f o r th e s id e s α ξ and β ξ
By t h is change o f v a r ia b le e q u a tio n (210) becomes:
(220)
I f we suppose t h a t V is a f u n c t io n o f a v a r ia b le ,
z
where μ
= μ
(S - S' )
is a fu n c tio n
(219) may be f u l f i l l e d .
( r - r ')
(221)
o f σ to be d e te rm in e d , the boundary c o n d itio n
I f e q u a tio n (220) a llo w s o f t h is change o f
v a r ia b le s , t h i s e q u a tio n becomes an o rd in a ry d i f f e r e n t i a l e q u a tio n .
T his tr a n s fo r m a tio n a lth o u g h n o t in g e n e ra l p o s s ib le , is p o s s ib le f o r
the ease o f B o y le 's law (20) and P o is s o n 's law (21)
For B o y le ’ s la w , e q u a tio n (201) a p p lie s , and
e q u a tio n
becomes
(222)
(220)
-7 5 -
I f we l e t z
= ((r-r')(s -s '))/4 a 2 t h is tra n s fo rm s in to
(223)
T h is is reduced to B e s s e l's e q u a tio n by th e t r a n s f o r m a t i o n 4
z= - x 2 ,
g iv in g
(224)
The s o lu t io n is th u s :
(225)
The s o lu t io n o f ou r problem is now com plete f o r the case o f B o y le 's
la w .
Fo r P o is s o n 's la w , by e q u a tio n ( 20 2 ), we may w r it e
(226)
where λ
is a c o n s ta n t.
E q u a tio n (220) th e n becomes:
(227)
I f we l e t z = -(((r-r')(s-s'))/((r+s)(r'+s'))) t h i s tra n s fo rm s to :
(228)
The s o lu t io n may be expressed in term s o f a h y p e rg e o m e tric s e rie s as
fo llo w s :
(229)
The s o lu t io n o f o u r problem is now com plete f o r th e case o f P o is s o n 's la w .
P re v io u s to the work o f Riemann, Ampere1 succeeded i n re d u c in g
the d i f f e r e n t i a l e q u a tio n s o f m o tio n to a li n e a r form when the gas
obeyed B o y le 's la w .
He uses q u a n t it ie s
s ig n from th e q u a n t it ie s r ,
s
α,β a n d η w h ich d i f f e r o n ly in
and w r e s p e c t iv e ly o f Riemann. The
e q u a tio n he o b ta in s is
(230)
Ampere, J o u rn a l de l'E c o le P o ly te c h n iq u e 1820 C a h ie r X V I I I , t Xi
, p 177
-7 6 w hich agrees e x a c tly w ith Riem ann's e q u a tio n (200) when m is g iv e n by
e q u a tio n (2 0 1 ).
The in t e g r a t io n o f th is
e q u a tio n was, however, Riem ann's
c o n t r ib u t io n .
Hadamard1 tra n s fo rm s e q u a tio n ( 141) by cha ng in g in de pe nd en t
v a r ia b le s from a , t to u,w and cha ng in g the dependent v a ria b le
a v a r ia b le
from x to
d e fin e d by:
z
= w a + u t - x
(231)
The tra n s fo rm e d e q u a tio n is :
(232)
He then tra n s fo rm s independent v a r ia b le s a g a in from u,w t o ξ,η
e q u i va le n t to the q u a n t it ie s -2S , 2 r o f Riemann.
is :
4(∂2
o
)-f({}=
z/ξη
w h ic h are
The r e s u lt in g e q u a tio n
(233)
where the f u n c tio n f is d e fin e d by :
(234)
T his equ a tio n is o f th e same f orm as e q u a tio n (200) o f Riemann and is
s o lv e d in an analagous manner.
b . A p p lic a tio n to
S p e c ia l Cases
1 . R iem ann's S p e c ia l Case2
A s p e c ia l case tr e a te d by Riemann is one in w h ich i n i t i a l l y
r and S have d i f f e re n t c o n s ta n t v a lu e s on e it h e r s id e o f a segment
αβ
o f th e x - a x is and i n t h is segment change w ith o u t a maximum p o in t
from th e one c o n s ta n t va lu e to th e o th e r . W
e assume:
(235)
Hadamard, lo c . c i t . pp . 162, 163
Riemann, lo c . c i t .
-7 7 -
T h is case is g iv e n by Weber1 and h is tre a tm e n t is here fo llo w e d . In
the re g io n α β ξ o f the x t p la n e (See F ig . 5
) r and S are found as
fu n c tio n s o f x , t by the method o f the p re v io u s s e c tio n .
O u tsid e o f
t h is re g io n we assume a t le a s t one o f th e q u a n t it ie s r and S to be
c o n s ta n t.
By com parison o f e q u a tio n s ( 168 ) and (169 ) w ith e q u a tio n
( 155) we see th a t in t h is case the wave is s in g le p ro g re s s iv e .
c u rv e α ξ ,
r =r1 and S decreases c o n tin u o u s ly w ith o u t maximum o r m in­
imum p o in ts from α
to ξ .
On the curve β
ξ , S = S2
c o n tin u o u s ly w ith o u t maximum o r minimum p o in ts from β to
eve ry p o in t u on the curve
angle
θ
On the
and r decreases
ξ
. From
αξ c o n s tr u c t a s t r a ig h t l i n e u u ' m aking an
f o r w hich
(236)
Then, a c c o rd in g to th e s o lu t io n f o r a s in g le p ro g re s s iv e wave u and ρ
are c o n s ta n t a lo n g e ve ry such l i n e .
β ξ
A lso fro m e v e ry p o in t v on th e curve
c o n s tr u c t a l i n e v v ' m aking an a n g le θ
c o tθ
f o r w h ich
= u + √(ϕ'(ρ))
Then a lo n g e ve ry such li n e u and ρ
(237)
are c o n s ta n t.
C o n s tru c t fo u r
s t r a ig h t l i n e s , α 1 , ξ 2 , ξ 3 , β 4 w ith a n g le s θ 1 , θ 2 , θ 3, θ4
such th a t
(238)
Where u' and ρ'
are th e va lu e s o f u and ρ a t th e p o in t ξ.
Then we assume:
(239)
. Weber, lo c . c i t .
pp 515-518
-7 8 -
The r e s t o f th e x - t - p la n e is f i l l e d
by the l i n e s μ μ' and v v'
lin e s d iv e rg e and hence th e s o lu t io n is everyw here u n iq u e .
. These
Under o th e r
assum ptions th a n (235) the se lin e s w ould n o t d iv e rg e and a d is c o n tin ­
u i t y would s e t in a t a c e r t a in tim e .
When α β is made to approach
zero the above s o lu t io n becomes Riem ann's second case o f an i n i t i a l
d is c o n t in u it y .
2. L in e a r Type Waves i n r andS
1
Lo rd R a y le ig h f in d s a s o lu t io n f o r th e case where r and S
are lin e a r such t h a t we have:
r = Ax+B ,
S = Cx +D
(240)
where A ,B ,C , and B a re fu n c tio n s o f t .
B o y le 's la w ,
(20) is assumed,
Fo r t h is case e q u a tio n s ( 176) and ( 177) become:
(241)
(242)
S u b s t it u t in g th e v a lu e s o f r and s from e q u a tio n s ( 240) we o b ta in :
(243)
(244)
Since th e se e q u a tio n s are i d e n t i t i e s
in x , the c o e f f ic ie n t s o f x in the
two e q u a tio n s must be zero and term s w ith o u t x must be z e ro .
We thu s
have fo u r e q u a tio n s to d e te rm in e the fo u r q u a n t it ie s A, B,C, and D.
Four a r b it r a r y c o n s ta n ts e n te r in t o th e e x p re s s io n s f o r these q u a n t it ie s
b u t two o f them m e re ly d e s ig n a te an a r b i t r a r y o r i g i n o f tim e and a r b it r a r y
o r ig in o f a b s d is e a .
We are n o t in te r e s t e d in the tr a n s fo r m a tio n o f axes
and hence we may express th e se q u a n t it ie s i n term s o f two a r b i t r a r y
c o n s ta n ts .
On s o lv in g f o r these q u a n t it ie s and s u b s t it u t in g t h e i r
v a lu e s in e q u a tio n s (240) we‘"o b t a in the f o llo w in g e x p re s s io n # f o r r and S:
L. R a y le ig h , P roceedings o f th e R oyal S o c ie ty , d u ly 8, 1910, p 256
-7 9 -
2 r = ( H + l)(x/t)) + ( H2 - 1) a lo g t + M
(245)
2S = ( H - l) (x/t)) + (H2 - l )
(246 )
a lo g t + N
where M+N=-2aH
Then a c c o rd in g to e q u a tio n s (f7/-2) we o b ta in f o r th e v e lo c it y and d e n s ity :
u = r-S = (x/t) - a H
a lo g
o rρ
(247)
ρ = r + S = H(x/t) + ( H2-1 ) a lo g t - 1 /2 (M+N)
= Ct(H^2)- 1 e(H
t)where C=eh
x/a
(248)
Thus f o r t h i s case, th e v e lo c it y rem ains alw ays li n e a r ,
re m a in in g c o n s ta n t in m agnitude a t th e o r i g i n .
wave becomes e ve r more g ra d u a l.
ponentia l d i s t r i b u t i o n .
I f H=± l ,
The d e n s ity a t any in s t a n t has an e x we see from e q u a tio n s (245) and (246)
t h a t e it h e r r o r S is c o n s ta n t and in the se
s in g le p ro g re s s iv e waves.
The s lo p e o f th e v e lo c it y
cases we f a l l back upon
I f H = o, u=x / t . and 1 /ρ is p ro p or t i o n a l to t .
The d e n s ity is c o n s ta n t f o r a g iv e n v a lu e o f t th ro u g h o u t th e gas b u t
decreases w ith tim e in such a way t h a t the s p e c if ic volume o f th e gas
is p r o p o r tio n a l to t .
The a c c e le r a tio n e xp e rie n ce d by a p a r t i c l e o f
gas is (∂w/∂t) + (u(∂u/∂x)) and by s u b s t it u t io n o f u from e q u a tio n (247 ) we
see th a t t h is is z e ro .
3 . L im ite d I n i t i a l D is tu rb a n c e 1
Riem ann's s o lu t io n do es n o t a p p ly to a re g io n
r o r S has a maximum.
in w h ich
N e v e rth e le s s , we can le a r n som ething o f the
p ro p a g a tio n o f a sound wave from an i n i t i a l d is tu rb a n c e c o n fin e d to a
re g io n
o u ts id e o f w h ich the gas is everywhere a t r e s t and has a con­
s ta n t d e n s ity j- 0
by a s tu d y o f e q u a tio n s (178) and (179) o r s ta te m e n ts
(180) and (1 8 1 ).
From th e se e q u a tio n s
we see t h a t r is p ro p a g a te d in
the p o s it iv e d ir e c t io n w ith a v e lo c it y
u+
the n e g a tiv e d ir e c t io n w it h a v e lo c it y
-u+
and S is p ro p a g a te d in
. A t a c e r t a in tim e ,
vsihic!
is In d e te rm in a te u n t i l th e e q u a tio n s are s o lv e d , th e h in d e r l i m i t o f the
r v a r ie s meets th e h in d e r l i m i t o f th e re g io n in w hich
re g io n in w hichAS v a rie s a f t e r w h ich thetw o re g io n s se p a ra te and i n ­
c lu d e between them a p o r t io n o f the f l u i d in i t s
1. R a y le ig h , T h e o ry o f
Sound, 1896, Y ol I I p 40 .
e q u ilib r iu m c o n d itio n .
-8 0 -
A f t e r the s e p a ra tio n o f th e two re g io n s , S is c o n s ta n t in th e gas to
the r i g h t o f the in te rm e d ia te re g io n and r is c o n s ta n t in th e gas to
th e l e f t o f t h is re g io n and hence th e re a re two s in g ly
p ro g re s s iv e
waves, one p ro pa ga te d i n th e p o s it iv e d i r e c t io n and one in th e
n e g a tiv e d ir e c t io n .
-8 1 II.
S p h e ric a l Waves o f F in i t e A m p litu d e
S p h e ric a l waves o f f i n i t e
been s tu d ie d .
tio n s
a m p litu d e do n o t seem to have
The case o f steady r a d ia l flo w is g iv e n by equa­
(9) and (10) w hich d e te rm in e u and ρ as fu n c tio n s o f r .
the im p ortan ce o f s p h e r ic a l waves in e xp e rim e n t
to in v e s tig a te somewhat th e case o f f i n i t e
waves.
A t any p o in t o f a f i n i t e
c o n d itio n s
From
we have been le d
a m p litu d e s p h e r ic a l
a m p litu d e wave
we may a p p ro xim a te
in th e im m ediate n e ig hb orh oo d o f t h is p o in t a t th e
in s ta n t by a stea dy r a d ia l flo w w ith superim posed in f i n i t e s i m a l
d is tu rb a n c e .
a.
T h is f a c t le d us to in v e s t ig a t e th e l a t t e r case.
I n f in it e s im a l D is tu rb a n c e Upon a Steady F lo w .
We must use e q u a tio n s (7 ) and (8 )
S ince by e q u a tio n (9)
r 2ρu is
where p = ϕ(ρ )[eq u a tio n (2 2 )]
c o n s ta n t f o r a steady flo w , in
our case we h a v e :r 2ρu = c + s
where c is c o n s ta n t and s is
in fin ite s im a l.
(249)
E q u a tio n (7) then
becomes: (250)
Now l e t s = (∂σ/∂t).
S u b s t it u t in g t h i s in th e above and in t e g r a t in g
we o b t a in : (251)
or
(252)
where R i s an a r b it r a r y f u n c tio n o f r .
S u b s t it u t in g the v a lu e s o f u and ρ fro m e q u a tio n s (249) a n d (252)
in equ a tio n ( 8 ) we o b t a in : -
D ropping h ig h e r powers o f i n f i n i t e s im a l s th a n th e f i r s t , w e o b t a in : -
-8 2 -
(253)
E x p re s s in g ϕ'((R -(∂σ/∂r))/r)
in th e powers o f ∂σ
/r
tha n the f i r s t in
by th e f i r s t two term s in th e exp an sio n
and a g a in d ro p p in g term s o f h ig h e r degree
(253) we h a v e :-
(254)
The term s in d e r iv a tiv e s o f
a re in f i n i t e s im a l s and hence the
term s n o t c o n ta in in g th e se d e r iv a tiv e s must be p la c e d eq ua l to z e ro .
Thus we ha ve: (255)
Thus we have an o rd in a ry d i f f e r e n t i a l e q u a tio n o f th e f i r s t o rd e r
to d e te rm in e R.
Then e q u a tio n (254) becomes: (256)
Let σ =
e
where P is a f u n c tio n o f r to be d e fin e d l a t e r .
Then e q u a tio n (256) becomes: (257)
T h is may be p u t in t o th e fo rm :
(258)
and e q u a tio n (258) becomes o f th e f i r s t o rd e r and ta k e s th e fo r m :(259)
T h is may be p u t in th e fo r m :(260)
-8 3 Let θ = ξ((η2ϕ'(η)-1)/(η2ϕ'(η))).
Then th e above e q u a tio n becom es:(261)
Now η = R / r 2 is known when R is d e te rm in e d fro m e q u a tio n (255)
and hence (261)
may be s o lv e d f o r θ as a f u n c tio n o f r .
W orking
back th ro u g h th e tra n s fo rm a tio n s made ,P th e n is known as a fu n c tio n
of r.
V alues o f 0~ a re th e n p ro p o g a te d w ith a v e lo c it y - P'
S ince e q u a tio n
a s o lu t io n .
.
(256) is l i n e a r , th e sum o f any number o f s o lu tio n s is
By choosing "a " com plex, CTbecomes a damped p e r io d ic
fu n c tio n .
b . F in i t e A m p litu d e Wave.
L e t r 2ρ= ∂F/∂r
and s u b s t it u t e in e q u a tio n (7)
.
Then in t e g r a t in g w ith re s p e c t to r we o b t a in : /t)+
(∂F
The fu n c tio n f ( t )
F u = f
(t)
(262)
may be re g a rd e d as in c o rp o ra te d in th e f u n c tio n F.
The r i g h t member o f e q u a tio n (259) may th e r e fo r e be re g a rd e d as z e ro .
U sing these v a lu e s o f u and P in e q u a tio n (8) we o b t a in : - I
(263)
I have n o t been a b le to s o lv e t h is e q u a tio n .
The s o lu t io n o f
the p re c e d in g case sh o u ld be a b le to be employed to b u ild up a
s o lu t io n f o r t h is case s im ila r to Lo rd R a y le ig h 's developm ent o f
P o is s o n 's in t e g r a l f o r a p la n e p o s it iv e p ro g re s s iv e wave as
d iscusse d im m e d ia te ly a f t e r e q u a tio n (1 5 9 ).
-8 4 E.
Plane Waves o f Permanent Regime W ith D is s ip a tiv e
F orces.
Waves o f perm anent regim e a re those f o r w hich th e
v e lo c it y wave does n o t change i t s
by e q u a tio n ( 1 6 7 ) ,t h a t t h is
fo rm .
We have a lre a d y seen,
is o n ly tr u e f o r a c e r t a in law o f p re s s u re .
T h is law can o n ly be m a in ta in e d by c o n d u c tio n o f h e a t o r by v is c o s it y
o r o th e r body fo rc e s a c tin g upon th e gas.
A p la n e wave o f
permanent regim e can be b ro u g h t to r e s t by a p p ly in g a c o n s ta n t
v e lo c it y to th e gas as a w h ole.
a case o f steady m o tio* n .
It,
t h e r e fo r e , may be tr e a te d as
The e q u a tio n s a p p lic a b le a re (3 ) and
(4) in w hich a c c o rd in g to s u b s t it u t io n ( 1 1 ), we s u b s t it u t e f o r
— —'Qih ,X — —
*
Assuming ste a d y m o tio n th e d e r iv a tiv e s
P ^X
P <*Y
w ith re s p e c t to t d is a p p e a r fro m the se e q u a tio n s and th e e q u a tio n s
become o r d in a r y .
E q u a tio n s (3) and (4 ) th u s become
pu = m
(264)
(265)
We s h a ll c o n fin e o u rs e lv e s to a ste a d y wave in w hich a t s u f f i c i e n t
d is ta n c e s in th e p o s it iv e and d ir e c t io n s , th e gas becomes o f a
u n ifo rm s t a t e .
I f th e fo r c e X is due to v is c o s it y ,
a t b o th te r m in a l s ta te s .
the te r m in a l
A ls o th e n th e re i s
s ta te and hence J d Q ,
i t va n ish e s
no h e a t c o n d u c tio n a t
is zero where Q, is th e h e a t
re c e iv e d by an elem ent o f gas m oving to a p o in t x , and where
the in t e g r a l is ta ke n from one te r m in a l s ta te to the o th e r .
I.
E n tro py C o n d itio n .11
From e q u a tio n s (264) and (265) we h a v e :(266)
1. R a y le ig h ,P ro c e e d in g s o f th e R oyal S o c ie t y , J u ly 8 , 1910,p p .258-260.
-8 5 The fo rc e o p e ra tiv e upon an elem ent o f mass
is X p <)^and the
t o t a l momentum g iv e n to the elem ent i s : (267)
A c c o rd in g to e q u a tio n s (264( and (266) i f
the in t e g r a t io n is take n
from th e i n i t i a l te r m in a l s ta te in w hich th e v e l o c i t y , d e n s ity
and p re s s u re are u ( , p
, and p f R e s p e c t iv e ly , to th e f i n a l te rm in a l
s ta te in w hich the se q u a n t it ie s have th e v a lu e s u , P ,and p ,
r e s p e c tiv e ly .
The p r i n c i p le o f c o n s e rv a tio n o f momentum
a p p lie s to th e v is c o s it y fo rc e s between th e p a r t ic le s o f the
gas and hence lx p < J t/in e q u a tio n (267) is z e ro .
T h e re fo re we have:
(268)
which is e q u iv a le n t to e q u a tio n (167) a p p lie d to te r m in a l s ta te s .
The energy expended by th e v is c o s it y fo rc e s is
From
e q u a tio n s (264) and(266) we h a v e :(269)
By e q u a tio n (264) th e r i g h t member is a fu n c tr o n o f p a n d p .
F or B o y le 's la w , (2 0 ), and P o is s o n 's a d ia b a tic la w ,
(21) , t h i s
q u a n tity is p o s it iv e o r n e g a tiv e a c c o rd in g as p ^ is le s s o r
g re a te r tha n p^ .
The c o n d itio n o f in c re a s e o f e n tro p y o r th e con­
d i t i o n o f d is s ip a t io n o f energy im p lie s
n e g a tiv e .
T h e re fo re , s in c e
t h a t t h is q u a n tity be
th e f i n a l p re s s u re exceeding th e i n ­
i t i a l p re s s u re denotes a wave o f c o n d e n s a tio n ,o n ly waves o f con­
d e n s a tio n are p o s s ib le .
II.
Heat C o nd uctio n.
Rankine s tu d ie s th e h e a t tr a n s fe r s necessary to m a in ta in
Earnshaw 's law o f p re s s u re ,
1. R ankine, P h il.
(167) i
T r a n s . , 1870, v o l.
He f in d s f o r a p e r f e c t gas
160, P a rt I I ,
p .2 7 7 .
-8 6 th a t to e f f e c t a change o f p re s s u re dϸ , a c c o rd in g to t h is la w ,
the q u a n tity o f h e a t added must b e :(270)
I n t e g r a t in g from the i n i t i a l to th e f i n a l s t a t e , we have s in c e
(271)
where p0 and v0 a re th e p re s s u re and s p e c if ic volume a t any p o in t
in the wave.
I f p0 , v 0 , a re i d e n t i f i e d w ith p 1 , v 1 , we o b t a in : (272)
S ince v 1 = 1/ρ1, by e q u a tio n (264) m v1 is eq ua l to u 1 .
M ul­
t i p l y i n g e q u a tio n (272) by v 1 we th e r e fo r e o b t a in : u 12= m
2 v12 =v1{((1/2)(y-1 )p 1 ) + ( ( 1 / 2 ) ( y + 1 ) p 2 ) } (273)
which is the square o f th e v e lo c it y o f p ro p o g a tio n w ith re s p e c t
to re g io n 1.
T h is is g re a te r th a n v e lo c it y o f sound o f i n ­
f in i t e s im a l a m p litu d e , f o r w hich u 12= y p 1,v 1 , f o r a c o n d e n sa tio n
and le s s f o r a r a r e f a c t io n , as we see from th e above fo rm u la .
The a b s o lu te te m p e ra tu re
is d e te rm in e d f o r a p e r f e c t
gas as f o llo w s : (274)
by the r e la t io n in e q u a tio n (2 7 1 ).
everywhere K(d
/x),
θ
The flo w o f h e a t is
where th e c o e f f i c ie n t o f h e a t c o n d u c tio n K may
be a fu n c tio n o f th e c o n d itio n o f th e gas.
from
I f we re c k o n Q
the i n i t i a l c o n d itio n o f c o n s ta n t p re s s u re p 1 ,
(275)
F or the d i s t r i b u t i o n o f p re s s u re Rankine f in d s from th e above
e q u a tio n :(276)
Under th e s u p p o s itio n t h a t k is c o n s ta n t, Rankine £±xdx
in te g r a te s e q u a tio n (276) and o b t a in s : -
-8 7 (277)
where
q = p - (1 /2) (p1 + p2) and q 1 = (l/ 2 ) (p 2- p1 ) ,
from th e p la c e where q e q u a ls z e ro .
the wave is i n f i n i t e l y
x b e in g measured
A lth o u g h m a th e m a tic a lly
lo n g , p r a c t i c a l l y th e t r a n s it io n is e ffe c te d in
a d is ta n c e comparable w ith
k /(m C (y + 1 )).
change s ig n , th e n u m era to r in
S ince
dx/d
p must n o t
(276) must be alw ays p o s it iv e
,and hence
p2/p1 must n o t exceed (y+1)/(3-y).
III.
V is c o s it y . 1
For v is c o s it y we must p la c e in e q u a tio n (265)
X = (4/3ρ)(d/dx(μ(du/dx))) a c c o rd in g to Lamb.2
In t h is ease a ls o R a n k in e 's c o n d itio n h o ld s ,
see e q u a tio n (268)
E q u a tio n (265) in te g r a te s i n t o : (278)
For B o y le 's law we f in d by i n t e g r a t io n : -
Here d x /d v never changes s ig n and a permanent wave o f co n d e n sa tio n
is
always p o s s ib le no m a tte r what the v a lu e o f the r a t i o p1/p2
may be.
The v e lo c it y o f p ro p a g a tio n in t o the r a r e r medium is :(279)
IV .
V is c o s ity and Heat C o n d u ctio n .3
Here the h e a t change in an elem ent c o n s is ts o f two p a r ts
dQ1 and dQ2 , th e fo rm e r the h e a t re c e iv e d by c o n d u c tio n and th e l a t t e r
t h a t developed in t e r n a lly under v is c o s it y .
F or
dQ1/dx and dQ
2/dx
we f in d th e e x p re s s io n s : (280)
1. R a y le ig h ,P ro c e e d in g s o f th e R oyal S o c ie ty ,J u ly 8 ,1 9 1 0 ,p p .269-271.
2. Lamb, Hydrodynamics ,1906, p p .53 5 ,5 3 8 ,5 3 9 .
3. R a y le ig h ,
lo c . c i t .
pp.271 -2 8 2 .
-8 8 U sing Rankines e q u a tio n (2 7 8 ), e q u a tio n (280 becom es:-
From t h is e q u a tio n i f we recko n Q1from th e te r m in a l s ta te v ,
(y - 1) Q = 1/2( y + l ) m(v1-V)(v-V2) +(4/3)m(μv(dv/dx)).
E q u a tio n (278) s t i l l a p p lie s and e x p re s s in g θ as p v / R , we may
o b ta in ,b y s u b s t it u t in g
θ
th e v a lu e o f p from e q u a tio n (278) ,
as a f u n c tio n o f v .
Then s u b s t it u t in g in the e q u a tio n o f c o n d u c tio n , we o b t a in : (282)
R a y le ig h assumes M a x w e ll's th e o ry as c o n n e c tin g th e s p e c if ic h e a t C,
a t c o n s ta n t
v o lu m e ,w ith th e c o e f f i c ie n t o f h e a t c o n d u c tio n K.
F or the th e o ry w hich assumes a m o le c u la r r e p u ls io n in v e r s e ly as the
o f the d is ta n c e ,
C a llin g t h is r a t i o h and w r i t i n g
f o r μ/m e q u a tio n (282) becom es:(283)
W r itin g
ξ
f o r v2 and U
f o r μ'(dξ/dx'), t h is e q u a tio n is reduced to
one o f f i r s t o rd e r, nam ely,
where f ( ξ ) = ((3(y+1))/8y)((√ξ1+√ξ2)/(√ξ3)) - (3/2) - h
and
F ( ξ ) = (3/4)h ( y + 1)(√ ξ 1
- √ ξ )(√ ξ
When U is determ ined as a f u n c tio n o f
- √ ξ 2 ).
from t h is e q u a tio n , x
found by sim p le in t e g r a t io n o f the e q u a tio n :-
R a y le ig h s o lv e s e q u a tio n (285) by a s e rie s o f a p p ro x im a tio n s .
is
-8 9 F.
Summary and O u tlo o k .
we have tr e a te d c o n tin u o u s and d is c o n tin u o u s waves.
We were a b le to t r e a t d is c o n tin u o u s waves in th re e d im e n sio n s.
The v e lo c it y o f p ro p a g a tio n o f a second o r h ig h e r o rd e r d is c o n t in u it y
w ith re s p e c t to th e gas is t h a t o f sound o f in f i n i t e s i m a l a m p li­
tu d e .
A lth o u g h th e re is much l i t e r a t u r e
on th e s u b je c t o f f i r s t
o rd e r d is c o n t in u it ie s i t appears t h a t such waves are
p o s s ib le t h e o r e t i c a ll y .
In r e a l i t y ,
even i f
p ro b a b ly n o t
th e v is c o s it y and h e a t
co n d u c tio n were everywhere e ls e n e g l ig ib l y , th e y would n o t be neg­
l i g i b l e a t th e d is c o n t in u it y and these f a c t o r s would p re v e n t th e
fo rm a tio n o r th e p e rs is te n c e o f th e d is c o n t in u it y .
S ince second
o r h ig h e r o rd e r d is c o n t in u it ie s a re pro p a g a te d w ith a v e lo c it y o f
th a t o f o p d in a ry sound w ith re s p e c t to th e gas, P o is s o n 's in t e g r a l
f o r a c o n tin u o u s wave h o ld s i f
th e v e lo c it y wave is d is c o n tin u o u s
in s lo p e o r c u rv a tu re , o r h ig h e r d e r iv a tiv e s o f th e v e lo c it y w ith
re s p e c t to x a re d is c o n tin u o u s .
I f th e re is a g iv e n i n i t i a l r e c ­
t i l i n e a r d is tu rb a n c e in a lim it e d r e g io n , th e curves f o r Riem ann's
v a r ia b le s r and s a re pro p a g a te d in th e p o s it iv e and n e g a tiv e
d ir e c t io n s , r e s p e c t iv e ly .
A f t e r these waves s e p a ra te we have
two s in g le p ro g re s s iv e waves, one in th e p o s it iv e and one in th e
n e g a tiv e d ir e c t io n .
The tim e f o r com plete s e p a ra tio n to ta ke p la c e
can be a p p ro x im a te ly d e te rm in e d .
The l i m i t a t i o n
in v o lv e d in
P o is s o n 's in t e g r a l, b y w hich a f t e r a c e r t a in tim e the v e lo c it y and
d e n s ity a re no lo n g e r u n iq u e ly d e te rm in e d , is r e a l l y v e ry s e rio u s .
The th e o ry o f sound waves o f f i n i t e
a p p lic a tio n in e x p lo s iv e waves.
a m p litu d e sho uld f in d i t s
Now v e l o c i t i e s o f e x p lo s iv e
waves have been measured w hich a re more tha n tw ic e th e v e lo c it y
-9 0 o f o rd in a ry sound.
L e t us suppose t h a t th e v e lo c it y o f a wave
is measured by the v e lo c it y o f p ro p a g a tio n o f th e maximum p o in t
o f the v e lo c it y c u rv e .
The maximum p o in t o f th e v e lo c it y curve
under t h is c o n d itio n would o v e rta k e th e f r o n t o f the wave w hich
o rd in a ry
moves w ith th e Av e lo c it y o f sound in a d is ta n c e le s s th a n th e wave
1
in
le n g th o f the wave. Hadamard s ta te s t h a t Ath e e xp e rim e n ts o f
p
V ie lle ^ on e x p lo s iv e waves, d is c o n t in u it y would ta ke p la c e in
a few c e n tim e te rs .
The in t r o d u c t io n o f
v is c o s it y and h e a t con­
d u c tio n would seem to be ne cessa ry to p re v e n t th e wave
becoming d is c o n tin u o u s .
c o n d u c tio n th a t have
from
The o n ly waves u n de r v is c o s it y and h e a t
been s u c c e s s fu lly tr e a te d are waves o f
permanent regim e in w hich th e gas passes from one u n ifo rm c o n d itio n
in f r o n t o f th e wave by c o n tin u o u s change w ith o u t maximum o r
minimum p o in ts to a n o th e r u n ifo rm c o n d itio n back o f th e wave.
We have seen t h a t a s in g le p ro g re s s iv e wave g e ne rated from a
lim it e d i n i t i a l d is tu rb a n c e must have a maximum p o in t .
To such
a wave the s o lu t io n f o r the presence o f v is c o s it y and h e a t con­
d u c tio n does n o t a p p ly .
We c o u ld i n f e r p ro b a b ly t h a t th e fo rw a rd
s lo p e o f the wave would a t t a i n a s ta te a p p ro x im a tin g a permanent
regim e and t h a t f o r th e backward s lo p e o f th e wave, P o is s o n 's i n t e ­
g r a l would rem ain a p p ro x im a te ly c o r r e c t and t h a t , in accordance
th e r e w ith , t h i s slo p e would become more g ra d u a l.
As to f u t u r e in v e s t ig a t io n , we need a g re a te r t h e o r e t ic a l
knowledge o f waves o th e r tha n p la n e waves.
fin ite
A s o lu t io n f o r a
a m p litu d e s p h e r ic a l wave would be v a lu a b le f o r such waves
can be more n e a rly ap p ro xim a te d in e xp e rim e n t tha n p la n e waves.
Waves, supposed to be p la n e , p ro p a g a te d in tu b e s , a re a ffe c te d by
1. Hadamard, lo c . c i t . p .1 8 5 .
2. V ie ll e , Comptes Rendus , 1898-99,
-9 1 the f r i c t i o n
o f the gas upon th e tu b e .
The
in flu e n c e o f v is c o s it y
and h e a t c o n d u c tio n on such waves a ls o needs to be s tu d ie d .
a ls o need a s tu d y o f th e phenomenon o f r e f l e c t i o n .
fir s t,
as the most sim p le c a s e ,fo r p la n e waves.
We
T h is can be s tu d ie d
B e fo re we can stu d y
r e f r a c t io n we must know more a b ou t waves o th e r tha n p la n e waves.
As to e x p e rim e n ta l v e r i f i c a t i o n ,
th e v e lo c it y o f p ro p a g a tio n
o f a wave in to a re g io n a t r e s t can be measured, and in so f a r as the
wave a p pro xim a te s a p la n e wave, th e d i f f e r e n t fo rm u la e f o r th e v e lo c it y
o f p ro p a g a tio n can be te s te d .
The re s is ta n c e o f the gas to th e m o tio n
o f a s o lid th ro u g h i t , i f we assume th e wave ge n e ra te d in f r o n t o f
the s o lid to be p la n e , can be used to d e te rm in e th e r e l a t io n between
the p re s s u re and th e v e lo c it y in a p la n e wave. R a y le ig h 1 assumes f o r
p r o je c t i le s
p r o je c tile
th a t the fo r e p a r t o f th e wave g e n e ra te d in f r o n t o f the
is one o f perm anent regim e under v is c o s it y and h e a t con­
d u c tio n and t h a t from th e v e lo c it y a s s o c ia te d w ith th e f i n a l s ta te o f
t h i s wave th e re is a second tra n s fo rm a tio n under w hich as the head
o f the p r o j e c t i l e
is approached th e gas changes a c c o rd in g to th e
a d ia b a tic law and th e v e lo c it y becomes th a t o f th e p r o j e c t i l e .
These
two tra n s fo rm a tio n s are supposed to ta k e p la c e in such a s m a ll d is ta n c e
th a t in sound ph oto grap hs o f r i f l e
b u lle t s ,e t c ,
th e y appear as one.
The d e te rm in a tio n o f th e shape o f th e d e n s ity wave does n o t le n d i t s e l f
r e a d ily to e x p e rim e n t.
The use o f a membrane w ith a d e v ic e to re c o rd
i t s movements and upon w hich th e wave im pinges is o f l i t t l e
the fr e e p e rio d o f the membrane and i t s
a v a il s in c e
i n e r t i a a f f e c t the r e s u lt s in a
manner which cannot be c a c c u ra te ly a llo w e d f o r .
I f §. wave c o u ld be m ain­
ta in e d steady in a tube b y - im p re s s in g a v e lo c it y upon th e gas as a w hole,
1. R a y le ig h ,P ro c . R oyal S o c .,J u ly 8, 1910, pp . 282-284.
-9 2 -
the shape o f the wave m ig h t be fo u n d .
on th e r e f le c t io n o f f i n i t e
around o b s ta c le s ,
P o s s ib le e xp e rim e n ts
a m p litu d e waves and t h e i r c u rv a tu re
w hich can be s tu d ie d by means o f sound p h o to ­
g ra p h s, a w a it f u r t h e r advances in th e o ry .