THE
JOURNAL
OF
ACOUSTICAL
THE
SOCIETY
Volume 23
OF
AMERICA
Number 4
JULY.
1951
Sound Scatteringby Solid Cylindersand Spheres*
IAME$ J. FARAN,JR.
A•oustic$ReseardtLaboralory,HarvardUniversity,Cambridge,
Massazhusetts
(ReceivedMarch 13, 1951)
The theoryof thescattering
ofplanewavesofsoundby isotropic
circularcylinders
andspheres
isextended
to take into accountthe shearwaveswhichcan exist (in additionto compressional
waves)in scatterersof
solidmaterial.The resultscan be expressed
in termsof scatteringfunctionsalreadytabulated.Scattering
patterns
computed
onthebasisof thetheoryareshown
to beingoodagreement
withexperimental
measurementsof the distribution-in-angle
of soundscatteredin water by metal cylinders.Rapid changeswith
frequency
in thedistribution-in-angle
of thescattered
soundandin thetotalscattered
energyarefoundto
occurnearfrequencies
of normalmodesof freevibrationof thescattering
body.
I. INTRODUCTION
can have a considerable effect on the distribution-in-
angleof the scatteredsoundand on the total scattered
HE scattering
of sound
wasfirstinvestigated
energy.Morse, with Lowan, Feshbach,a.ndLax, later
mathematicallyby Lord Rayleigh.t However,
because
of thecomplexity
of themathematical
solution, extended his solution to include the effects of comwavesinside(fluid)cylindricaland spherical
heonlyconsidered
thelimitingcasewherethescatterers pressional
a These resultsare also given in convenient
are smallcomparedwith the wavelength.The solution scatterers.
whose
forscattering
by rigid,immovable
circularcylinders
and form in termsof severaladditionalphase-angles
values
are
tabulated.
The
object
of
the
research
reported
spheres,
notnecessarily
smallcompared
with the waveby cylinders
and
length,was givenin convenientform by Morse,who herehasbeento studysoundscattering
of solidmaterial(whichwill supportshearwaves
definedand tabulatedvaluesof phase-angles
associated spheres
waves).The mathematical
withthepartialscattered
waves,in orderto simplifythe in additionto compressional
solution
will
be
given
first,
after which experimental
complicated
dependence
on besselfunctions.
• Although
most solidscatterersin air can be consideredrigid and apparatusand resultswill be described.
immovable,it is valid only in a few specialcasesto
II. THE MATHEMATICAL
SOLUTION
assumethat a scattererin a liquid mediumis rigid and
List of Symbols
immovable.In general,the soundwaveswhichpenetrate the scatterermust be taken into account,as they
Most of the symbolsusedhereare,in the appropriate
sections
of the analysis,the sameas thoseusedby Love
* Thispapercontains
theessential
results
of a thesissubmitted
to the Faculty of Harvard Universityin partial fulfillmentof the
and thoseusedby Morse:
requirements
for thedegree
of Doctorof Philosophy.
Thisresearch
hasbeenaidedby fundsmadeavailableundera contractwith the a
ONR.
' LordRayleigh,TheTheoryofSound(DoverPublications,
New
= radiusof cylinderor sphere;
a •, b•, c• = expansioncoefficients;
a Mathematical Tables Project and M.I.T. Underwater Sound
Laboratory, Scatteringand Radiationpom Circular Cylindersand
Company,New York, 1936),first edition,and (1948), second Spheres(U.S. Navy Department, Office of Research and Inedition.
ventions, Washington, D.C., 1946).
York, 1945), first Americanedition.
• P.M. Morse, Vibra//on and Sound (McGraw-Hill Book
405
406
A
Az
JAMES
J.
FARAN,
= vectordisplacement
potential;
= z-component
of vectorpotential;
= 4)-component
of vectorpotential;
=velocity of compressional
waves in the
JR.
cylinderalongthe negativex-axis,as shownin Fig. 1.
As in the solutionsgiven previouslyfor rigid and fluid
scatterers)-a the wave motion external to the scatterer
is assumedto consistof the incidentplanewaveand an
outgoingscatteredwave. It is desiredto find the
amplitudeof the scatteredwave as measuredat large
= velocityof soundin thefluidsurrounding
the distancesfrom the cylinder.The mathematicalexpresscatterer;
sions for displacementand dilatation inside and for
pressureand displacementoutside the cylinder will be
= Young'smodulus;
=(-1)•;
found in generalform first, after which the application
of the properboundaryconditionsat the surfaceof the
= sphericalbesselfunctionof the first kind;
= besselfunction of the first kind;
cylinderwill lead directlyto the solution.
=•o/c•;
The wavesinsidethe cylinderwill be represented
by
= ,o/c2;
suitable solutionsof the equation of motion of a solid
= ,o/c3;
elasticmedium, which may be written4
= order integer;
(x+ 2)VA-(2a) = pOu/Ot
= sphericalbesselfunctionof the secondkind
scatterer;
C2
E
J
j•( )
ld )
kl
k2
ka
n,,( )
)
P
pi
= velocity
ofshear
waves
in thescatterer;
= besselfunction of the secondkind;
= pressure;
= pressurein incidentwave;
= pressurein scatteredwave;
P,,(cosO)= Legendrepolynomial;
P0
= amplitudeof pressure
in incidentwave;
= cylindricalcoordinates;
r, O,g
= sphericalcoordinates;
r, O,4
[rr], [-rO],[rz] = stresscomponents
in cylindricalcoordinates;
['rr], [rO], [r•] = stresscomponents
in sphericalcoordinates;
t
= time;
u
ur, uo
u•.•
= displacement;
u•.,
=radial componentof displacement
in scat-
x, y, z
= rectangular
coordinates;
xo_
x3
= k2a;
= kaa;
= components
of displacement
in the solid;
=radial componentof displacementin incident wave;
tered wave;
Fro. 1. Choiceof coordinateaxesfor scatteringby cylinders.
where
a•, fi,,,/L,,/5,/, •, •= scatteringphase-angles;
A
•
= dilatation;
=Neumann factor; •0= 1; •=2,
a = V. u
2a=VXu.
n>0;
X, •
= Lam• elasticconstants;
2•
ot
= rotation;
= density of the scatterer;
Oa
= densityofthefluidsurrounding
thescatterer;
•
= Poisson'sratio;
• •
,I•
= boundaryimpedancescatteringphase-angle;
= scalardisplacementpotential;
•o
= angularfrequency(2•rf).
(2)
and
From Eq. (1) can be derivedthe equations,
V"a= (m/X+ 2•) O2a/Otø-
(3)
w(2a) = (pt/la)Oø-(2&)/Ol
(4)
and
which define the wave velocities
c•= E(x-l-2u)/p,3•= [E(1- a)/m(lq-•)(1-
2,)3 • (5)
and
Scattering by Solid Circular Cylinders
c2= (•/p,)i= [-E/2m(lq-.)-]t.
(6)
Plane wavesof soundof frequency•o/2r in a fluid
medium are incident upon an infinitely long circular Solutionsof Eq. (1)can be foundby assumingthat the
cylinder
of someisotropic
solidmaterial.Let theaxisof
• A. E. H. Love, A Treatiseon the MathematicalTheory of
the cylindercoincide
with the z-axisof a rectangular Elasticity
(Dover Publications,New York, 1944), fourth edition,
coordinate
system,andlet theplanewaveapproachthe
p. 141.
SCATTERING
BY
CYLINDERS
AND
SPHERES
401
fluid medium,whichcan be
displacement
canbederivedfroma scalaranda vector tion for a (nonviscous)
written
potential:
u-- -
vx A.
(7)
Up = (11c•')O'pl
The displacement
thuscan be thoughtof as the sumof The incidentplanewaveis represented
bys
two displacements,
one associatedwith compressional
Pi= Poexp(--jkax)= Poexp(--jkar cos0)
waves and the other with shear waves. If we assume that
the potentialssatisfythe equations,
= PoE
cosn0.
(lS)
•-•0
V• = (1/c,90•'I•/Ot2
(S)
and
•A = (1/ce)OaA/ot
•
(9)
The radial componentof displacement
associated
with
this wave is
we c•n showthat A(= V- (-- V•)) satisfiesEq. (3), and
ui.,= (1/ pao•a)Opd
Or
that 2fi(=VXVXA) satisfiesEq. (4). That •e•
a•umptionsdo lead to a valid solutionof Eq. (1) may
be •en by noting that the solutionswe shall obtain
satisfyEq. (1) by directsubstitution.
If wenowchange
to a •lindrical coordinatesystemdefinedby
=
e,(--j)'--J,(kar)
pao•
• .=o
cosn0.
(16)
dr
The outgoingscatteredwave must be symmetrical
x= r cos0, y= r sin0, : = z,
about 0= 0 and therefore of the form
it can be seenthat pressureand disp•ment mustbe
s•metri•l about 0=0 (the directionoI the positive
x-•xis). Moreover,becausethe cylinderis of h•ite
length,and the incidentplanewaveof infiniteextent,
of displacement
associated
with
there can be no dependence
on z, and it is logicalto The radialcomponent
this wave is
•ssumethat there is no displacementin the -.-direction.
Subjectto the• conditions,the solutionof Eq. (8) •n
1
d
p.=
cosn0.
be written
•= • a•J•(kff)
cosn0.
pa• a •-a
(10)
•0
dr
The factors c, are the unknown coefficientswhich must
(The timedependence
factorexp(j•t) will beunde•tood
in all theexpressions
repre•ntingwaves.)Examination
of Eq. (9) showsthat, subjectto the conditionsdiscus•d above,•e vectorpotentel c•n have no componenth •e r- or the O-direction.
The vectorEq. (9)
then reducesto a sca•r equationin A •, and its solution
can be written
A,=• b•J•(k•r)
sinn0.
be evaluated.
The followingboundaryconditions
areappliedat the
surface
ofthecylinder:(I) Thepressure
in thefluidmust
be equalto the normalcomponent
of stressin the solid
at the interface;(II) the normal(radial) component
of
displacement
of the fluid mustbe equalto the normal
component
of displacement
of thesolidat theinterface;
and (III) the tangentialcomponents
of shearingstress
(11) mustvanishat the surfaceof the solid.That is,
Only sine terms appearhere, becau• the vector potenthl must be anti-symmetricalabout 0=0 in order
that the displacementderived from it shall be symmetricalabout0=0. Now, by Eqs. (7) and (2),
pi+p.=--[rr]
at r=a,
(19)
(20)
and
at
r=a.
(21)
In cylindricalcoordinates,
e
• I-ha.
d
Err]= xa+ 2•audOr: 2me•E(•/1- 2.)a+ OudOr-],
[,03 = ,[ (1/,) (audoo)+ fie/a,) (.40 ],
1
,,=
and
(13)
[rs]=
OudOa+ Oud Or].
and
A=k•• •.•a.J,(k•r)
cosn0.
By the conditions
of symmetry,[-rz]=0 everywhere.
fromEqs.(15),(17),(14),(12),(16),
(14) Uponsubstitution
(18),and (13), theboundary
condition
Eqs.(19), (20),
The wavesin the fluid surrounding
the cylinderwill
be representedby suitablesolutionsof the wave equa-
• Seereference
2, secondedition,p. 347.
• See reference4, p. 288.
408
JAMES
J.
FARAN,
JR.
and (21) become,for the nth mode,
wave,is definedby
XlJnt(Xl)a•-- nJ ,•(x2)b,•
tann• = tan• •(xa)
X Etan••+ tanot.(xa)•[tan••+ tantt,(x3-].
+
09a)
-
(2oa)
The intermediatescatteringphase-angles
• n(x)= tan-•E- J•(x)/N •(x)-l,
a•(x) = tan-•E- xJ ,' (x)/ J n(x)-l,
and
•(•) = tan-'['-- •,%' (•)/-V•(•)3,
and
havebeendefined
andtheirvaluestabulatedpreviously.
a
The angle •b., which is a measureof the boundary
impedance
at the surfaceof the scatterer,is given,for a
Solvingthe• equationssimultaneously
for c. is labori- solidscatterer,by
ousbut straightfo•rd. The resultis
tan•= (- p•/m) tan•(x•, a),
(23)
-v0..(-j)
wherethe newscattering
phase-angle
•,(xt, a) is given
whe• •., the pha•-•fft
angle of the nth scattered by
•,(x•, •)= tan-•
•
xxJ•'(x•)
2n2J•(x
]
•
2 (a/l--2a)x•:[J•(x•)--J•"(xx)•
- 2n•Ex:••)3
x•J•t(x•)-- J•(x•)
neJ•(x•)--x•Jn(x•)+x•J•
I'
(24)
(x2)J
For convenience
• computing
valuesof thisfunction,it canbe writtenin terns of •e anglea•(x):
tana•(xx)
n•
r•(xt,
a)=tan
-t x22t•a•(xt)+•
[•a•(.•n•--Ix22
.
2tana•(xx)+n=--}x•
• n•[tan•(x•)+13
[
Although•'naswrittenaboveis explicitlya functionof
xx and.x2,it can be considered
a functionof xx and a,
sincethe ratio of x• and x• is a functionof v only.
Valuesof •'•(x•, a) computedfromEq. (25) for a= « are
givenin TableI. Forconvenience
in findingthetangent,
the valueof the anglelying between4-90ø is givenin
(25)
the table.The dottedlinesindicatethat •(x•, a) passes
through4-90ø betweenthe adjacententries,and thus
serveto pointout the infinitiesof tanl',(xb a). It will be
seenbelowthat the infinitiesof tanl%(xx,•) occurat
preciselythe frequenciesof thosenormal modesof free
vibration of the scattererwhichsatisfythe conditions
of symmetryof the scatteringproblem.The dottedlines
in Table I thus mark the locations of the normal modes
of vibration of the scatterer. For other values of Poisson's
ratio the functionswill be similar,the only difference
being shiftsin the locationsof the normal modes.
The scatteringpattern, or distribution-in-angle
of
pressurein the scatteredwave at largedistancesfrom
the cylinder,can be foundfrom Eqs. (17) and (22),
usingthe asymptoticexpressions
for the besselfunctions
for large arguments:
I
2 \«
•
Ip.l•,.•,Pota.-•r)
l•oe,•sin.,•exp(j•,•)cos
(26)
Scattering by Solid Spheres
INCIDENT PLANE
WAVE
Fro. 2. Choiceof coordinateaxesfor scatteringby spheres.
Let us assumethat planewavesof soundin a fluid
mediumare incidentupona sphereof someisotropic
SCATTERING
TA•
xt
0.0
0.2
0.4
n=O
n=l
0.00ø
1.37
6.27
BY
0.00ø
3.59
15.47
AND
409
SPHERES
I. Valuesof l'dxt, a) for the cylindricalcasefor a= «.
n=2
- 45.00ø
--44.13
-43.33
CYLINDERS
n=3
0.00ø
2.06
8.73
n=4
n=5
0.00ø
1.52
6.33
n=6
0.00ø
1.20
4.97
n=7
0.00ø
1.01
4.02
0.130
ø
0.86
3.46
n=8
n=9
0.(1t3
ø
0.76
3.03
0.00ø
0.67
2.68
0.6
0.8
14.35
25.60
-41.18
-37.48
36.24
60.28
20.10
35.26
14.07
25.40
11.15
19.90
9.12
16.24
7.80
14.00
6.80
12.12
6.04
10.24
1.0
38.90
-31.24
79.03
51.42
37.95
30.52
25.18
21.23
18.52
16.84
1.2
52.22
- 18.06
' ' •g•'.•i
65.40
50.79
41133
34.96
30.31
26.58
23.56
1.4
63.81
-t-62.49
-79.71
76.03
61.87
52.07
45.11
38.96
34.61
31.22
1.6
73.08
"•'1'.•;•
- 74.29
SS.8S
70.71
61.11
53.83
47.93
42.73
39.11
1.8
80.33
-68.94
89.73
77.52
68.61
61.51
55.74
50.71
46.45
82.84
87.06
74.55
79.25
67.96
73.05
62.24
67.88
57.38
63.03
59.43
2.0
2.2
' ' •'.•õ
2.4
2.6
-85.16
--81.30
2.8
-7.53
-63.54
-56.21
-81.57
53.17
64.93
-49.71
-37.81
-77.67
-73.08
-89.43
-86.35
83.05
86.22
77.21
80.71
72.25
76.15
67.97
72.17
64.10
68.51
-77.32
73.30
- 19.43
-64.48
-83.42
88.92
83.58
79.25
75.52
72.15
3.0
-72.71
79.69
+6.92
+68.10
-80.28
-88.67
86.02
81.83
78.34
75.20
3.2
-66.65
85.83
34.29
-74.64
-76.17
-86.39
88.15
84.09
80.71
77.75
3.4
3.6
-57.35
--40.29
74.49
87.38
54.21
67.25
-65.75
-55.95
-68.06
-11.94
-84.05
-81.35
--89.92
-88.10
86.04
87.77
82.78
84.58
79.94
81.86
3.8
-6.92
76.96
-40.24
+85.24
-77.55
--86.28
89.35
86.18
83.57
' ' •)'.•
-11.45
' ' •'.•õ
-69.80
-84.30
' ' •b'.i•i
87.59
85.03
-{-70.33
+28.36
-78.33
-27.54
-81.87
-87.69
88.91
86.34
4.0
86.05 +15.39
36.63
-89.02
-t-34.15 -86.10
4.2
58.25
4.4
70.32 -80.06
80.59
4.8
81.68 -70.05
87.52 "'1¾.•
5.0
84.94
89.87
4.6
77.19
-83.24
' ' •'.õ•
- 76.03
- 60.60
84.69
57.79 -73.27
78.81
68.66
53.06
+75.02 -78.22
--86.15 ' ' •b'.•õ
- 84.38
- 88.63
88.64
-56.75
' ' •5'.•
-29.41
-82.10
-87.38
89.69
- 34.99
- 82.97
+ 70.97
- 78.49
- 86.02
- 66.94
88.55
- 70.46
87.54
- 89.28
Pressure
in the incidentwaveisrepresented
by7
solidmaterial.Let the centerof the spherecoincidewith
theoriginofa rectangular
coordinate
system,
andletthe
planewavesapproach
thesphere
alongthenegative
z axis,asshown
in Fig.2. Theanalysis
isverysimilarto
thatfor thecylindrical
case.We transfer
to spherical
pl= Poexp(--jkaz)=Poexp(-jkar cos0)
=P0• (2n+
1)(-j)"j•(kar)P,(cosO).
coordinates
definedby
x= r sin0cos4, y= r sin0sin4, z= r cos0.
The outgoing
scattered
wavewill be of theform,
p,=• c,[-j,(kar)-jn•(kar)•P,(cosO
(27)
Because
theincidentwaveapproaches
alongtheaxisof
4, thereis nodependence
on4. It is logicalto assume
The sameboundaryconditions
at the surfaceof the
that there is no component
of displacement
in the
scatterer
are
applied
to
the
expressions
fordisplacement,
4-direction,
and it followsthat the onlynon-zero
compressure,
and
dilatation,
which
are
either
givenabove
ponent
of thevector
potential
in thiscaseisAo.The
potentials
arethenfoundto beof theforms,
•-- • a.j.(k,r)P•(cosO)
or derivablefrom the above.In sphericalcoordinates
the stresscomponentsare
[rr-I=XA+2uOu,/ar=
2OlC==[-(a/1-2a),X+audOr-],
[rO]=/•[Ou-•ø--uø+-I
OU•
l,
and
d
A,=,,=o
•'.b•j.(k=r)--P.(cosO).
dO
L Or
r
r O0_l
Seereference2, secondedition,p. 354.
410
JAMES J. FARAN, JR.
and
The intermediate
angles,
[r•]=• sin0
ao• Or
.
• •(x)= tan-•[- j • (x)/n• (x)],
a•(x)= tan-•[- xj•'(x)/j•(x)],
/•• (x)= tan-• [- xn•' (x)/n• (x)],
By carryingtheanalysis
throughasin the cylindrical
case,we find that
c•= -- Po(2n+1)(--j) •+asinn• exp(jn•),
where the phase-shift•
definedby
havebeendefinedand their valuestabulatedprevi(28) ously.
8 The boundaryimpedance
phase-angle
av• is
of the nth scatteredwave is
tanebb=-(oa/ox)
tanf•(x•,•),
tan,/•= tan•(xa)[ tan• •-btana•(xa)]/
(29)
wherethenewscattering
phase
anglef•(xx,½)isgiven
tan•+
•(xx,,)=tan -•
definedby
tan/g•(xa). by
x•j•'(xO
2(n2+n)jn(x•
]
2(,/1-2v)x•[-j•(xO-jn"(xO]
2(n•+n)[j•(x•)_x••,(x•)]
•
x2•
x•j•'(xO-j•(xO (n•+n-2)j•(x•)+x•j•"(x•)
--
--
x•j•'(xO- j •(xO
•
.
(n•+ n- 2)j • (x•)+ x••j ,,"(x•)]
Thisfunction
canbeexpressed
in termsof theanglea•(x):
r•(x•, a)= tan-•
x••
tana•(x0
n•+n
tana•(x0+1 n•+n-l-•x•+tana•(x•)
••
.
(3O)
2n•+n-•x•+2
tana•(x0
(n•+n)[tana•(x•)+l]
I
tana•(x0+ 1
n•+n- 1- •x•" tana•(x•)d
Values
ofthisfunction
computed
fromEq. (30)for• = « repetition
rate,served
to identifyinterfering
pulses
aregiveninTableII. Thedottedlinesagainindicate
the which,
stillreverberating
in thetankfromtheprevious
infinitiesof tan•(x•, •), that is, the normalmodesof transmitted
pulse,happened
to arriveat thereceiver
at
free vibration of the scatterer.
the sametime as the pulseto be measured.
A small
The distribution
in angleof pressure
in thescattered adjustment
of the average
pulserepetition
ratewas
waveat largedistances
fromthe sphereis foundfrom effective
in controlling
interferenc
e of thistype.Both
Eqs.(27)and(28)bymeans
oftheasymptotic
expres-transducers
employed
x-cutquartzcrystals
operated
at
sions
forthespherical
bessel
functions
forlargeargu- resonance.
Serious
distortion
of the short(64 t•sec)
pulses
bythetransducers
wasprevented
bylowering
the
P0 o,
Q of the quartzcrystals
by increasing
the radiation
Thiswasaccomplished
by inserting
between
[P•
I ,--•ar
I•o(2n+
1)sin•
exp(j•)P•(cos0)l.
(31)loading.
the crystals
andthe wateran acoustic
quarter-wave
transformer
in theformofa thindiskofPlexiglas.
The
HI. EXPERIMENTAL APPARATUS
ments:
amplitude
of thescattered
soundpulses
wasmeasured
Measurements
of thedistribution-in-angle
of sound byamodified
substitution
method,
anoscilloscope
being
scattered
in waterby metalcylinders
weremadefor the usedas an indicator.
Thepulses
werebrought
to a
purposeof checkingthe theory.Thesemeasurementsstandard
deflection
ontheoscilloscope,
changes
in the
weremadein a largesteeltankat or neara frequencypulse
amplitude
beingcompensated
by changes
in the
of onemegacycle
persecond.
A sound
projector
in one attenuation
in the receiving
system.
end of the tank irradiatedthe scattererwith sound.A
receiving
hydrophone
wasmounted
in sucha waythat IV. CONIPARISONOF THEORY AND EXPERIMENT
it could
easily
bemoved
toanyposition
lyingona circle
TheeXPerimental
datawere
normalized
sothatthey
concentric
withthescatterer,
andservedto measure
the
could
be
compared
with
scattering
patterns
computed
distribution
in angleof the pressure
in the scattered
from
the
theory.
In
order
to
do
this,
the
amplitude
of
wave.Shortwavetrainsor "pulses"
of sound
wereused
intheincident
wave
(P0)was
measured
by
in orderthatthemeasurement
of eachpulsecouldbe thepressure
moving
the
receiving
transducer
to
the
position
of
the
effectivelycompleted
beforesoundreflectedfrom the
3.Caremust
betakentodistinguish
between
the
wallsofthetankcould
reach
thereceiving
hydrophone.• Seereference
andspherical
cases,
since
thesame
symbols
areused
for
A novelfeature,
frequency
modulation
of thepulse cylindrical
thescattering
phase-angles
in bothcases.
SCATTERING
BY
CYLINDERS
AND
SPHERES
411
TABLEII. Valuesof rn(xi, a) for thespherical
casefor a= «.
0.0
0.00ø
0.2
0.4
0.6
03
1.16
4.66
10.58
18.89
-- 45.00'
--44.72
--43.57
--41.62
--38.54
0.00ø
3.05
12.77
29.97
51.40
0.00ø
1.94
7.77
17.88
31.55
0.00ø
1.41
5.91
13.38
23.47
0.00ø
1.13
4.66
10.38
18.83
0.96
3.91
8.73
15.93
0.00ø
0.81
3.33
7.50
13.58
0.00ø
0.00o
0.73
2.93
6.58
12.19
0.00o
0.64
2.62
5.88
11.01
1.0
29.12
-33.76
70.41
46.79
35.44
28.67
24.25
20.74
18.32
16.09
1.2
40.28
--26.18
83.89
60.52
47.88
39.29
33.22
29.67
25.84
23.32
1.4
51.05
-- 12.80
' ' •5•
71.44
58.67
50.00
43.23
38.34
34.49 31.51
1.6
1.8
60.55
68.41
+16.87
89.15
--80.31
-- 75.00
79.64
85.78
67.68
74.68
59.02
66.42
52.20
59.69
46.57
54.53
42.41
49.80
2.0
74.78
"•'5'.õ•
- 70.20
"•'•.•
80.14
72.50
66.17
60.82
56.28 52.29
2.2
2.4
79.92
84.14
--65.27
--59.50
--85.58
--82.18
84.46
87.97
77.29
81.17
71.50
75.75
66.61
71.13
62.27
67.13
58.30
63.16
2.6
87.71
48.97
"•'.õ•
-- 78.94
"•'9'.$)
84.39
79.27
74.89
71.11
67.61
2.8
' ' •'9'.iõ
61.45
--41.61
--75.41
--86.44
87.06
82.18
78.10
74.62 71.07
3.0
-- 86.24
70.09
-- 25.03
-- 70.56
-- 83.95
89.38
84.63
80.80
77.41
74.43
3.2
--83.36
76.48
--0.51
--55.48
--81.37
"•'•.õ•
86.75
83.03
79.87
77.00
3.4
-80.31
81.81
+27.37
' ' •'l'.õi
--78.27
--86.56
88.61
84.98
81.93 79.23
3.6
-- 76.72
88.64
49.05
-- 68.79
-- 73.44
-- 84.57
"•)
86.69
83.75
3.8
-- 72.00
63.24
--59.85
--59.37
--82.35
--88.10
88.21
85.33 82.88
4.0
--64.84
85.64
72.99
--47.06
+65.07
--79.48
--86.50
89.61
86.71
84.30
4.2
--51.66
89.00
81.77
--24.13
"•i
--74.70
--84.81
' ' •'•.•
87.99
85.62
4.4
4.6
--22.39
+26.20
56.58
' ' •'8'.,•õ ' ' •5•'.i•- +13.38
--86.12 +73.10
47.64
--83.70
80.80
--80.54
--75.60
--60.64
+48.49
--82.81
--80.07
--87.76
--86.41
--84.90
--88.62
88.89
5.0
70.03
--80.89
84.41
--62.36
' ' •'8'.i•
--60.25
--83.03
--87.50
89.85
4.8
+9.84
30.55
"k•'.ii
68.81
--70.16
scatterer.After normalization,
it wasstill necessary
to
adda factoramounting
to 1.9dbto theamplitudeof the
scatteredsoundin orderto bringthe experimental
data
into goodagreementwith the theory.This correction
factorhasbeenexplained,
and its valuecomputedwith
goodaccuracy,by takinginto accountthe fact that the
illuminationof the scatterervariesin phaseandamplitude alongits length2
The part of Eq. (26) which was evaluatedin computingthe patternswas
}1 ,• sin•exp(jr/n)
cosnOl,
and the corresponding
numericalscaleis shownon all
the patterns usedas illustrations. The values of Poisson's
84.65
--75.21
38.72
45.77
81.23
89.17 86.81
' '•.)i
87.88
the values of Young'smoduluswere measured(to
within4-5 percent)by findingthefrequency
of thefirst
modeof fiexualvibration of the cylindricalspecimen
mounted so that it could vibrate as a fixed-free bar. The
valueof • wasthendeterminedby meansof Eq. (5). In
somecaseswherethe patternwasvery sensitiveto frequency,it wasnecessary
to choose
a valueof • slightly
different from that based on the Young's modulus
measurementin order to bring the measuredand
computed
patternsintoagreement.
Comparison
of the
valueof ¾oung's
modulus
corresponding
to theassumed
value of x• with the measuredvalue servesin thesecases
to indicatethe degreeof agreement
betweenexperiment
and theory.
Figures3 through13 are measuredand computed
scatteringpatterns for cylindersof various sizes.The
ratio for the variousscattererswere assumed,because
of the difficultyof measuringthis constantdirectly;but pressurein the scatteredwaveis plottedlinearlyagainst
scatteringangle.In eachcasethe arrow indicatesthe
• J. J. Faran, Jr., Sound Scatteringby Solid Cylindersand directionof the incidentsound.The angle0 is measured
Spheres,Technical Memorandum No. 22 (March 15, 1951), from the top centerof the graph,the incidentsound
AcousticsResearchLaboratory,Harvard University,Cambridge,
Massachusetts.
comingfrom the direction0= 180ø. For eachsize of
412
JAMES
J.
FARAN,
JR.
.
½
•
in.wavescatteredby a
Fxo. 3. Scatteringpattern for brasscylinder0.0322 in. in Fro. 5. Computedamplitudeof pressure
rigid, immovablecylinderfor xa= 1.7.
diameterat 1.00me/see.Points:Measuredamplitudeof pressure
in the scatteredwave.The measured¾oung'smoduluswas 10.1
X 10n dynes/cm
a. Cur•e:Computedpatternfor xa= 1.7, xt--0.6,
(Fig. 6) is somewhatunusual;the amplitudeof sound
scatteredback in the directionof the sourceis nearly
soundis fully
scatterer,the pattern computedon the basisthat the zero.This near-nullin the back-scattered
explained
by
the
mathematical
solution
in which the
scattereris rigid and immovableis includedfor comparison.
Figures3 and4 showscatteringpatternsfor brassand
steel (drill rod) cylindersof the samesize,for eachof
whichxa--1.7. Thesepatternsare both very similarto
that for a rigid, immovablescattererof the samesize
•r--«, m=8.5 g/eraa.
(Fig.5).
Figures6-8 showscatteringpatternsfor cylindersof
variousmaterialstwice as large in diameter,that is,
x.•=3.4. The pattern for a brasscylinderof this size
'
.75
FIO.4.Scattering
pattern
fors•edcylinder
0.0042
in.indiameter Fro.6. Scattering
pattern
forbrass
cylinder
0.0625
in. indiameat 1.00inc/s/to.
Points:Measured
amplitude
of pressure
in the terat 1.02mc/sec.
Points:
Measur"•it-amplitude
ofpressure
in the
scatteredwave. The measured¾oung'smoduluswas 20.0X 10u
dynes/cm
a. Cume:Computedpattern.•forlxa=l.7, xt=0.45,
*
0.28, m--=7.7g/cmL
scatteredwave. The measured¾oung'smoduluswas 10.4X 10u
dynes/cm
a. Curae:Computedpattern for xa-$.4, x•=1.185,
a--l, pt=8.5 g/cms (corresponding
to E--10.SX10n dynes/cron).
SCATTERING
BY
CYLINDERS
AND
SPHERES
413
term for n--2 in the seriesfor the scatteringpattern
suddenlybecomes
very largein amplitudeand of the
properphaseto cancelthe sumof all the othertermsat
0= 180ø.This, in turn, is broughtaboutby the presence
of an infinity in the tan•(x•, a) functionfor a•-« at
xt--1.18-.- (corresponding
to a normalmodeor resonanceofthescatterer),in theneighborhood
of whichthis
functiongoesrapidly througha wide rangeof values
causingthe variationsin the coefficientof the n=2
term. The value of x• for the computedpattern of
Fig. 6 waschosen
to givea deepnotchat 0= 180ø,and
the frequencyat which the experimentalpattern was
FIG. 8. Scatteringpatternfor steelcylinder0.0625in. in diame-
ter at 1.00inc/sec.Points:Measu'h'T•
amplitude
of pressure
in the
scattered wave. The measuredYoung's moduluswas 19.5X 10•t
dynes/cm
:. Curue:Computedpatternfor x:= 3.4,xt= 0.9, •r= 0.28,
Ot= 7.7 g/cm•.
'•-•
Figures10, 11, and 12 are scatteringpatternsfor
brass,steel,and aluminumcylindersfor whichxa= 5.0.
Fro. 7. Scattering
patternfor coppercylinder
0.0625in. in
diameter
at 1.00inc/sec.
Points:Measured
amplitude
of pressure
in the scatteredwave. The measured¾oung'smoduluswas
11.9X10
n dynes/cm
2. Curve:Computed
patternfor xa-3.4,
a (corresponding
to E=12.YX10
n
x•-- 1.08, *--•,•' m=8.9 g/cm
dynes/cm2).
measuredwas chosenthe sameway. Figure 7 is the
scattering
patternfor a coppercylinderof the same
size.The value of x• for the coppercylinderis near
enough
to 1.18that the coefficient
of then=2 termis
stilllarge,butin thiscaseit isoftheopposite
phase
and
causesthe soundscatteredin the direction0= 180ø to be
somewhat
largerin amplitudethanthat scattered
by a
rigid,immovable
cylinderof thissize(Fig. 9). The
velocity
ofsound
in steelissomuchhigherthanthatin
brassor copperthat this scattererbehaves
nearlyas
though
it wererigidandimmovable,
anditsscattering
pattern(Fig.8) islittledifferent
fromthatforthe.rigid, Fro. 9. Computedamplitudeof pressurein wave scatteredby a
immovable
case.
rigid, immovablecylinderfor x,=3.4.
414
JAMES
J.
FARAN,
JR.
of xl than uponthe densityof the scatterer.The pattern
for a rigid,immovablecylinderof thesamesizeisshown
in Fig. 13,andit is apparentthat all thesepatternsfor
metal cylindersof this size bear little resemblance
to
this limiting case.
The theorythusverifiesthe existenceof nullsin the
back-scatteredsoundfor cylindersof variousmetals,
andat the properfrequencies;
but a furthertestis to see
whetherit predictsproperlythe mannerin which the
amplitudeof theback-scattered
sound(andtheshapeof
the entirepattern) changes
with frequency.In orderto
testthis,patternsweYemeasured
for thebrasscylinder
of Fig. 6 and the steelcylinderof Fig. 11 at two other
frequencies,
3 percentbelowandabovethat at whichthe
referencepatternswere measured.The corresponding
patternspredictedby the theory were computedby
makinga corresponding
changein the valuesof the x
parameters.In Fig. 14, the pattern of Fig. 6 is reproducedin the center,and thosefor 3 percentchangesin
frequencyare shown at either side. In Fig. 15, the
pattern of Fig. 11 is reproducedin the center,and the
patternsfor 3 percentchanges
in frequencyareshownon
either side.The theoryis seento predict the changesin
the measured
patternswith gratifyingprecision.
These
groupsof patternsalsoemphasizethe fact that the null
Fla. 10.Scatteringpatternfor brasscylinder0.093in. in diameter at 1.015inc/sec.Points:Measuredamplitudeofpressure
in the
scatteredwave.The measured
Young'smoduluswas 10.0X10n
dynes/cm
2. Curve:Computedpatternfor xa= 5.0, x• 1.78, •-«,
p•-8.5 g/cma (corresponding
to E• 10.2X10tl dynes/cm2).
The frequencyof measurementof the pattern of the
brassscattererwaschosento give the deepestnotchat
120ø, and the value of xx was*chosento make the
patterns agree. The choiceof the value of xx is well
substantiatedby the measurementof the Young's
modulus of this scatterer, since the value of E correspondingto the chosenvalue oœx• is within 2 percentof
the measuredvalue.Figure 11 showsthat, just as in the
caseof brass(Fig. 4), thereis a near-nullin the sound
back-scattered
from a steelcylinderat a frequencynear
that of the lowest-frequency
normal modewhich, for
a= 0.28, occursat xt= 1.30---. Figure 12 showsthat the
same is true of an aluminum scatterer of the same size.
Althoughthe velocityof compressional
wavesin steelis
not the sameasthat in aluminum,the valuesof Poisson's
ratio differ sufficientlythat this normal modeoccursin
thesetwo materialsfor the samephysicalsize of the
scatterers.Thesetwo patternsare so similarthat they
are seento dependmuchmorecriticallyuponthe value
Fro.11.Scattering
pattern•or
steel
cylinder
•.09375
in.in
diameterat 0.99 inc/sec.Points:Measuredamplitudeof pressure
in the scattered wave. The measured ¾oung's modulus was
19.3X10n dynes/cm
2. Curve:Computedpattern for xa=5.0,
x•- 1.293,,- 0.28, p•= 7.7 g/cma(corresponding
to ϥ 19.7X10n
dynes/cma).
'- -'
,
SCATTERING
BY
CYLINDERS
AND
SPHERES
415
in the back-scatteredsound is very sensitiveto
ßfrequency.
Measurements
of scatteringby a few sphereswere
madewith thisapparatus.However,because
the sound
scatteredby a spheredivergesin three dimensions
(insteadof two, as in the caseof a longcylinder),the
measurement
wasfoundto be very difficult,becauseof
the reducedmargin of signal to noise.The measurements(andalsocomputations)
indicatethat, although
rapidchanges
in the patterndo occur,thereis no null in
the soundback-scattered
in water by a brasssphere,
near its lowest-frequency
normalmodeof vibration.
V. REMARKS
ON THE BEHAVIOR
SCATTERERS
OF SOLID
It is interestingto examinethe behaviorof certainof
the functionswhich appear in the mathematicalsolution, especiallythe tan•'n(ggl,
o') functions.As noted
above, it can be shown that the infinities of the
tan•(x•, a) functionsoccurat preciselythe frequencies
of thosenormalmodesof freevibrationof the scattering
bodywhichsatisfythe conditions
of symmetryof the
scatteringproblem. This can be done by applying
boundaryconditions
to expressions
for displacement
and
dilatationwritten in generalform in termsof an unknownfrequency.The boundaryconditions,for free
vibrations,are simplythat the normalcomponentof
stressand the tangentialcomponents
of shearingstress
Fro. 13. Computedamplitudeof pressure
in wavescatteredby a
rigid,immovablecylinderfor xa= 5.0.
at thesurfaceof thebodymustbothvanish.Solvingthe
resultantequationfor frequency(in termsof unknown
x• and x2parameters)
givesa condition
which,in the
cylindrical
case,is identical
to requiring
the denominatorof Eq. (24)to vanish.
toFor a--«, thefirstfewof
thesenormalmodesoccurat the followingvaluesof the
frequencyparameter:
for n=O, x•=2.17.--,
5.43.--,
8.60.--;
for n=l, x•--1.43-.., 3.5•7..., 3.74..-;
for n=2, xt=l.18..-,
2.25.--, 3.98..-;
for n=3, xt-l.81...,
3.01-.-, 4.65---;
for n=4, x•=2.36-.-,
etc.
Thefirstnormalmodes
forn= 1,2,and3 occurforlower
valuesof x• (lower frequencies)
than that for n=O,
contrary
towhatwemightexpect.
Thereason
forthisis
that there are no shear waves associatedwith the n= 0
normalmodes.The complicated
wavestructurewhich
comprises
a normalmodecanbe realizedat a much
lower frequencywith shearwavesthan without, becausethevelocityof shearwavesis somuchlowerthan
Fro. 12. Scatteringpatternfor aluminumcylinder0.0925in. in that of compressional
waves.
diameterat 1.00 nic/sec.Points:Measuredamplitudeof pressure
For fluid scatterers,the functionstan•(x•,
in the scattered wave. The measured ¾oung's modulus was
7.0X10n dynes/cm
a. Curve:Computedpattern for
xt•l.17, a-a, ,ot=2.7 g/era (corresponding
to E 7.2X10n
dynes/cron).
a) in
Eqs.(23)and(29'•arereplaced
s by thefunctions
l0For detailsof this demonstration,
seereference9.
416
JAMES
J.
FARAN,
JR.
Fro. 14.The scatterlag
patternof Fig. 6 repeated
for comparison
with measured
andcomputed
patternsfor frequencies
3 percenthigher(right) and 3 percentlower(left).
tanot•(x•).It is interestingto notethat the infinitiesof surroundingfluid, however, this frequencyshift is
thesefunctionsalsocorrespond
to frequencies
of normal usuallysmall.
modesof free vibration of the (fluid) scatterer,sincethe
While measurements
were being made with the exinfinitiesof tanas(x•) occurat the zerosof J,•(xt) or perimental apparatus at frequenciesnear that of a
js(x•), inthecylindrical
andspherical
cases,
respectively. normalmode,it was in somecasespossibleto observe
The coefficientcs in the seriesfor the scatteringpa.t- "ringing"of that normalmodefollowingthe endof the
tern doesnot attain its maximumvalue at exactly the pulse;that is, longtransientscouldbe observedat the
frequencies
of the normalmodesof freevibrationof the end (andat the beginning)of the scatteredpulse.By
scatterer.Sincethe amplitudeof cs is proportionalto adjustingthefrequency
to givethemaximumamplitude
sin•, cs reachesits maximumvalue when tan• be- of the transientat the end of the pulse,it was thus
comesinfinite. This representsa shift in the resonant possibleto measurethe frequencies
of variousnormal
frequency
ofthenormalmode,andthisshiftisattributed modes.It wasalsopossible
to identifytheordern of the
to the reactivecomponentof the acousticimpedance excitedmode,becausethe amplitudeof the transient
presentedto the scattererby the surroundingfluid, i.e., followingthe pulse was proportionalto cosn0.These
the reactivecomponentof the radiationloading.In the transients were not noticeable in the case of the first
caseof solidshavingdensitiesgreaterthan that of the normal mode for n=2. Apparentlythe dampingby
2'0
Fro. 15. The scatteringpattern of Fig. 11 repeatedfor comparisonwith measuredand computedpatternsfor frequencies
3 percenthigher (right) and 3 percentlower (left).
SCATTERING
BY
CYLINDERS
AND
radiationintothe waterwasgreatenoughto causeany
ringingto die out quickly.However,the first normal
2.2
modesfor n=O, 1, a•nd3 were observedand identified
2.0
for brassand steelcylindricalscatterers
of appropriate
sizesand showedgoodagreementwith the frequencies
predictedby the theory.
That thereare sizeableshiftsin the frequencies
of the
normalmodeswith changes
in Poisson's
ratio suggests
that finding the frequenciesof one or more of these
normalmodesof vibrationmightprovidea methodof
measuringPoisson'sratio for cylindricalor spherical
specimens.The variation of the frequenciesof these
normal modes with
Poisson's ratio
is illustrated
1.6
1.4
1.2
in
of the
second
normalmodefor n = 2 isalsoshownin the graph
for the sphericalcase.In this connection,as well as in
the scatteringproblemitself, the potential utility of
havingthe l',(xt, a) functionscomputedfor a wide
rangeof valuesof Poisson'sratio will be evident. A
computationprogramto yield theseresultsappearsto
bejustified.The frequencies
of the normalmodescannot
becomputed
explicitly,but canbefoundeasilyfromthe
locationsof the infinitiesof the tant,(x•, a) functions.
It is interestingto comparethe behaviorof the taM,
functionsfor solid and fluid scatterersas xt, the frequencyparameterfor the scatterer,approaches
zero.
For solidscatterers,
eithercylindricalor spherical,as
tan•-•O,
n•l;
417
1.8
Figs. 16and 17, wherethe valuesof x• at whichthe first
normalmodesfor n=O, 1, 2, 3, and 4 occurare plotted
as functions of Poisson's ratio. The variation
SPHERES
tan•pr•pa/pl;
1'(•4 26 28
.30 .3Z .34
PO•SON'S RATIO•
Fro. 16. The valuesof xt for the first few symmetricalnormal
modesof freevibrationof a solidcylinderplottedas functionsof
Poisson's ratio.
the fluid becomeincompressible;
but as this happens,
the scattererdoesnot necessarily
becomerigidto shear
distortions.It must then be that, for n= 2 and higher,
shapedistortionsof the incompressible
fluid scatterer
make the componentsof the scatteredwave different
from what they wouldbe if the scattererwere rigid.
Because
thefluidscattererneverbecomes
rigidasx•--}0,
one can only passfrom this solutionto the caseof the
rigid,immovablescattererby lettingthedensitybecome
infinite.
Two summarycomments
canbe addedregardingthe
general
features
of scattering
by solidcylinders
and
spheres.
If the frequencyof the incidentsoundis lower
than that of the first symmetricalnormalmodeof free
vibrationof the solidscatterer,and if the densityof the
t•nq• = (- os/m) tans •(x]),
scattereris greaterthan that of the liquid, thereis little
as xr•O,
differencebetweenthe scatteringpattern for the solid
scattererand that for a rigid,immovablescatterer.But,
In neithercase,by lettingx•-•O, do we realizethe case rapidchanges
in theshapeof the scattering
patternand
of therigid,immovable
scattererwheretan•= 0 for all in the total scatteredpower(or scatteringcrosssection)
n. In orderthat xx= toa/ct--•0at finitefrequencies
in the
solidcase,the velocitiesof both the compressional
and
shearwavesmustbecomeinfinite,andthescattererdoes
2.6
indeedbecomerigid.The only term wheretan• does
while for fluid scatterers,where
not vanish is that for n= 1. This deviation from the
rigid,immovable
caseissimplydueto oscillation
of the
scattereras a wholein synchronismwith the incident
soundfield.Thus,by settingx]=0 in the solutiongiven
herefor solidscatterers,we can calculatethe scattering
froma rigid,movable
cylinderor sphereof densitym. To
passto the caseof the rigid,immovablescatterer,we
mustalsorequirethat the densityof the scatterer
2.2
.1.8
1.4
come infinite. In the case 6f a fluid scatterer, as
onlytan•0approaches
thevaluefor thelimitingcaseof
a rigid,immovable
scatterer.For n= I, tanq>•behaves
in the sameway asin the caseof the solidscatterer,and
represents
oscillationof the scattererin synchronism
with the incident sound. Now, for fluid scatterers,in
orderthat xr-•0 at finite frequencies,
it is necessary
that
O.6
.24
.26
.28
.•0
.•.
.•,4
;56
POISSON'SRATIO•
Fro. 17. The valuesof xt for the first few gymmetrlcalnormal
modesof free vibration of a solidsphereplotted as functionsof
Poisson's ratio.
418
W.
J.
CUNNINGHAM
VI.
canoccurwithsmallchanges
in frequency
in thevicinity
of certain of the normal modes of free vibration of the
ACKNOWLEDGMENTS
The author is indebted to Professor F. V. Hunt for his
solidscatterer.Thesechanges
includethe appearance
of guidance
andencouragement
throughout
thisinvestigadeepminiresin the scatteringpatternat certainangles tion. The assistanceof Dorothea Greene, who perand may include,for cylinders,
a near-nullin the sound formedthe laboriouscomputations
for Figs. 16 and 17,
scattered back toward the source.
is gratefullyacknowledged.
THE JOURNAL
OF THE ACOUSTICAL
SOCIETY
OF AMERICA
The Growth
VOLUME
of Subharmonic
23, NUMBER
4
JULY,
1951
Oscillations
W. 3- Cv•mmoa•
Yale University,lgewHaven,Connecticut
(ReceivedJanuary14, 1951)
Subharmonic
oscillations
at one-halfthefrequency
of excitation
mayappearin certaintypesof oscillating
systems,
amongwhichis thedirect-radiator
loudspeaker.
Theseoscillations
occurat verynearlytheresonant
frequency
of thesystemwhentheparameters
of thesystemaremadeto var• at twicethisfrequency.
The
rate of growthof the subharmonlc
dependsuponthe amountof variationof the parametersreiativeto the
dissipation
in the system.If the dissipation
is small,the rate of growthmay belarge.In the loudspeaker,
conditions
aresuchthat the rateof growthis usuallysmallfor typicalconditions
of operation.
a singledegreeof freedom.a?For the subharmonicto
HE
generation
of
subharmonic
by
a appear,
direct-radiator
loudspeaker
hasoscillations
often been
obthe quiescent
resonantfrequencyof the system
served3
-• Suchoscillations
usuallyoccurat one-halfthe must be very nearly one-halfthe excitingfrequency.
frequencyof the currentsuppliedto the loudspeaker, Further, operationmust be suchthat underexcitation
andappearfor onlycertaindiscretefrequencies
nearthe
center of the audio spectrum. In most cases, the
subharmonicis not presentunlessthe loudspeakeris
beingoperatednearits maximumpower.Whenpresent,
the subharmonicis easily audible, even though sound
pressure
measurements
indicatethe amplitudeof the
subharmonic
isonlya fewpercentrelativeto thefunda-
the resonantfrequencyof the systemis causedto vary
at the excitingfrequency.This variation must take
placein sucha way that sufficient
energyis beingsuppliedto thesystemto replacethat lostby dissipation.
If
morethanthisamountof energyis supplied,
theamplitudeof thesubharmonic
grows,in theory,withoutlimit.
Ultimately,in practicalsystems,
someadditionaleffect
mental. The statement has been made that this sub- takesoverandtheamplitude
achieves
a steadyvalue.
harmonicdistortionis usually of litfie practicalimIn orderto givea simpleexampleof this type of
in some
portancein the operationof the loudspeaker.
• The operation,an electriccircuitwill be considered
reasoningis basedon the observedfact that an ap- detail. This circuit contains in series combination an
preciable
lengthof timeis requiredfor theamplitudeof inductance
L, a resistance
R, and a capacitance
C. If q
the subharmonicto grow to its ultimate value. Since is the instantaneous
chargeon the capacitance,
the sum
typical programmaterialis of constantlychanging of voltagesaround the circuit is
nature, thereis little opportunityfor the subharmonic
rA+ R+q/C=O,
(1)
to build up. In the followingrather simplediscussion,
the growthof the subharmonic
oscillation
is consideredwheredotsindicatetime derivatives.In someway the
with the intent of determiningwhat factorz influencethe capacite;nce
is made to vary sinusoiclally
in time by an
rate of growthand why this rate is low for the loud- amount AC about the mean value Co.The instantaneous
speaker.
capacitanceis
Subharmonicoscillationat one-halfthe frequencyof
C= Co(l+a sin2•o,t),
(2)
an excitingforcemayoccurin oscillating
systems
having
wherethe angularfrequencyof the variationis taken as
• H. F. Olson,AcousticalEngineering(D. Van NostrandCorn~ 20•,,andamAC/Co.Evidentlya canneverexceed
unity.
pany, Inc., New York, 1947),p. 167.
It
is
possible
to
show
that
such
a
variation
in
capaciap.O. Pealerson,
J. Acoust.Soc.Am. 6, 227-238 (1935), and 7,
64-7O (1935).
tancecan add energyto the oscillatingcircuit.The
a F. yon Schmoller,TelefunkenZeitung67, 47-54 (June,1934). resonant
angularfrequency
of thecircuitin itsquiescent
4 O. Schaffstein,
Hochfrequenztechn.
Elektroakust.45, 204-213
(1935).
sN. Minorsky,NonlinearMechanics
(EdwardsBrothers,Inc.,
sSeereference
2. Also,H. S. Knowies,"Loudspeakers
and room Ann Arbor, 1947), Chap. XIX.
acoustics," Sec. 22, Henney's Radio Engineering Handbook
7N. W. McLaehlan,OrdinaryNonlinearDifferentialEquations
(McGraw-HillBookCompany,Inc., New York, 1941),p. 902.
(OxfordUniversityPress,London,1950),Chap.VII.