LECTURE 39
Sound waves (continued)
Our governing equations are now just the x-comp equation of motion,
Du
1 "p
=!
.
Dt
! "x
(1)
and mass conservation (in which thermo energy equation for adiabatic motion, ideal
gas law, and definition of potential temp knocked ln ! out in favor of ln p ).
1 D ln p !u
+
=0 .
! Dt
!x
(10)
Linearizing the governing equations
Consider a reference atmosphere (atmosphere before wave arrives) in which u =
U (constant), v = w = 0, density is constant, !c , and pressure is hydrostatic, P(z).
In presence of the wave we write:
u(x,t) =U + u !(x,t) ,
!(x,t) = !c + ! !(x,t) ,
p(x,t) = P(z)+ p !(x,t) ,
where a prime denotes a perturbation variable. Perturbations are small in sense that:
u ! << U , ! ! << !c , p ! << P . Expand out Du/Dt in x-eqn of motion (1), get:
!u
!u
1 !p
+u
="
!t
!x
! !x
Plug in u =U + u ! , p = P + p! , etc
%
(*
''
!U !u #
!U
!u #
!U
!u #
1
!p # ****
'' !P
+
+U
+U
+ u#
+ u#
= $
+
!t
!t
!x
!x
!x
!x
!x ****
!c + ! # ''' !x
''& "
*)
"
"
"
"
0
since U is const
0
since U is const
0
since U is const
use Binomial approx
0
since P is indep of x
from Binomial approx:
!
1
=
!c + ! !
(1
! %''
1 "$
!
1 "$
!! %
$$1 + ' )
$$1 ( '''
=
"
!c $# !c ''&
! ! % !c #$ !c ''&
!c $$$1 + ''''
!c '&
$#
1
"u #
"u #
"u #
1 %
! # ( " p#
+U
+ u#
= $ '''1 $ ****
"t
"x
"x
!c '& !c *) "x
Now linearize this equation: throw out the non-linear terms (products of
perturbations). These products are really tiny (since the perturbations are tiny) so we
can safely neglect them. The resulting equation is linear.
!
"u #
"u #
1 !p "
+U
=$
"t
"x
!c !x
!
#"
&
%% +U " ((u ) = * 1 " p ) .
%$ "t
"x (('
!c "x
(11)
Now work with (10). Set it up for easy linearization by reverting from lnp back to p:
1 1 Dp !u
+
=0
! p Dt !x
mult by ! p
Dp
!u
+!p
=0
Dt
!x
Expand out total deriv
!p
!p
!u
+u
+!p
=0
!t
!x
!x
Plug in u =U + u ! , p = P + p! , etc
$
&
')
& !U
!P ! p#
!P
! p#
!P
! p#
!u # ))))
+
+U
+U
+ u#
+ u#
+ ! (P + p#) &&&
+
=0
&& !x
!t
!t
!x
!x
!x
!x
!x ))))
&&% "
)(
"
"
"
0
0
0
0
P is indep of t
P is indep of x
P is indep of x
!p "
!p "
!p "
!u "
+U
+ u!
+ ! (P + p !)
=0
!t
!x
!x
!x
2
U is const
Now linearize it: throw out the non-linear terms
! p"
! p"
!u "
+U
+!P
= 0.
!t
!x
!x
!!
$
## +U ! && p ' + ! P (u ' = 0 .
#" !t
!x &&%
(x
(12)
Wave solutions of the linearized governing equations
Equations (11), (12) are 2 equations in 2 unknowns ( p! , u ! ). Combine them to
get 1 equation for just p! or u ! . Okay, lets get rid of u ! . In (12) u ! shows up only as
!u "/!x . We want to "create" a similar term in (11). So take !/!x (11):
2
"!
%
$$ +U ! '' !u ( = ) 1 ! p (
$# !t
!x ''& !x
!c !x 2
Now plug in
"!
%
$$ +U ! '' p (
"!
% $
!x ''&
1 !2 p (
$$ +U ! '' # !t
=
)
$# !t
!x ''&
)!P
"c !x 2
!u "
from (12) (watch minus sign)
!x
Mult by !!P and rearrange
2
"!
%2
$$ +U ! '' p ( ) !P ! p ( = 0 , Wave equation for sound waves
$# !t
!x ''&
"c !x 2
(13)
where we've used special "symbolic" notation for the first term:
! 2
2
2 $
"!
%2
#
&#
&
$$ +U ! '' p ! = %% " +U " ((%% " +U " (( p ! = ## " + 2U ! +U 2 ! &&& p ' .
## 2
%$ "t
$# !t
!x ''&
"x (('%$ "t
"x (('
!x!t
!x 2 &&%
" !t
Consider a trial solution of (13) as a sine or cosine (your choice). Okay, work with
p ! = Acos(kx ! !t) .
(14)
3
Plug (14) into (13). Pre-calculate some of the derivatives:
! p"
= #Ak sin(kx # !t)
!x
!2p "
= !Ak 2 cos(kx ! !t)
2
!x
!p "
= !A(!!)sin(kx ! !t) = A! sin(kx ! !t)
!t
!2p "
= !A!2 cos(kx ! !t)
2
!t
!2p "
= Ak ! cos(kx ! !t)
!x!t
So (13) becomes (careful with signs!):
!A! 2 cos(kx ! !t) + 2UAk ! cos(kx ! !t)!U 2Ak 2 cos(kx ! !t)
"P 2
+
Ak cos(kx ! !t) = 0
#c
Divide through by common factor of Acos(kx ! !t) :
!!2 + 2Uk ! !U 2k 2 +
!P 2
k = 0.
!c
2
"!
! %'' !
$
Combine first 3 terms (they came from $$ +U
p so we're just collapsing them
!x ''&
# !t
back to their original form):
2
!(! !Uk ) +
!P 2
k = 0.
!c
To find ! , rearrange above equation and take square root.
2
(! !Uk )
=
"P 2
k
#c
4
! !Uk = ± k
! = kU ± k
!P
"c
!P
"c
Relating P/ !c to reference temp T through ideal gas law, P = !c RT , ! becomes
! = kU ± k !RT
Dispersion relation for sound waves.
(15)
So, p!=Acos(kx"!t) is a solution of the wave equation provided ! satisfies (15).
A dispersion relation tells us how ! varies with k and the physical parameters
for a particular wave type (sound waves in this case). Dispersion relations are derived
from dynamical and thermodynamical equations. In contrast, ! = ck or equivalently
c = ! /k , arose from kinematic considerations, by analyzing the generic form of the
wave phase. They are not dispersion relations.
Dispersion relations describe (implicitly) how waves of different wavenumber
propagate with respect to each other. To make that explicit, use dispersion relation in
c = ! /k to get c. For sound waves:
c =U ± !RT .
(16)
No k in (16), so waves propagate with same phase speed regardless of wavelength.
The dispersion relation tells us that sound waves are non-dispersive. [Good thing,
too. If this wasn't the case, humans couldn't use sound to communicate.]
We can rewrite (16) as
c =U ±cs
where cs ! !RT ,
(17)
5
is adiabatic speed of sound. Under ordinary conditions at sea level, cs ! 350 m s"1 .
The ± ambiguity arises because 1D waves can propagate away from source
toward right ( + x) or left ( ! x). U can also be + or ! . Consider the frequency
! = k (U ± !RT ) heard by an observer under different conditions:
If wind is westerly (U > 0) and observer is to right of source, observer hears
frequency of the rightward moving wave +k !RT but increased by a positive
Doppler shift kU. An equivalent situation occurs if there is no wind but the sound
source (e.g., an ambulance) moves to the right at speed U, i.e., toward observer.
If wind is easterly (U < 0) and observer is still to right of source, observer hears
frequency of the rightward moving wave +k !RT but decreased by a negative
Doppler shift kU (negative since U < 0). An equivalent situation occurs if there is no
wind but the sound source moves to the left, i.e., away from observer.
6