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11. Acoustic waves and shocks
Reading: Shu, Vol. 2, Ch. 15
11.1 Acoustic waves of low amplitude
Let us consider an adiabatic (or isentropic
s
=const.) fuid in equilibrium at rest, in which the
pressure will vary in coordination with the density
P
∝
ρ
γ
γ
=
c
p
c
V
= ratio oF speci±c heats
(11
.
1)
The pressure gradient thereFore is
~
∇
P
=
∂P
∂ρ
±
±
±
±
s
~
∇
ρ
=
c
2
s
~
∇
ρ
(11
.
2)
with the adiabatic speed oF sound
c
s
=
c
s,
0
ρ
ρ
0
!
(
γ

1)
/
2
(11
.
3)
where
ρ
0
and
c
s,
0
=
∂P
∂ρ
±
±
±
±
s,ρ
0
are the uniForm density and sound speed.
Let us For simplicity consider only one spatial dimension.
Noting that the adiabatic relation
11.1 replaces the energy equation we write the remaining hydrodynamical equations as
∂ρ
∂t
+
u
∂ρ
∂x
+
ρ
∂u
∂x
= 0
(11
.
4)
∂u
∂t
+
u
∂u
∂x
=

1
ρ
∂P
∂x
=

c
2
s
ρ
∂ρ
∂x
(11
.
5)
IF we assume that the fuid quantities are only minimally perturbed,
ρ
=
ρ
0
+
ρ
1
u
=
u
1
P
=
P
0
+
P
1
(11
.
6)
we can write down perturbation equations containing only the terms that are linear in the
perturbations.
∂ρ
1
∂t
+
ρ
0
∂u
1
∂x
= 0
(11
.
7)
∂u
1
∂t
=

c
2
s,
0
ρ
∂ρ
1
∂x
(11
.
8)
and combining these two equations
∂
2
ρ
1
∂t
2
=
c
2
s,
0
∂
2
ρ
∂x
2
⇒
ρ
1
=
f
(
x

c
s,
0
t
) +
g
(
x
+
c
s,
0
t
)
(11
.
9)
1
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View Full Document The perturbations propagate with the sound speed an do not change their shape!
This is
equivalent to saying that in the lowamplitude limit sound waves do not show dispersion,
ω
is
not a function of
k
and the phase velocity of sound waves is constant.
Does that still hold for sound waves of Fnite amplitude?
Equation 11.3 tells us that can no
longer set the adiabatic sound speed constant, so some dispersion would occur.
This is the
result of the nonlinear nature of the hydrodynamical equations.
On should note that wave
damping, e.g. by viscous friction, adds other complications to the dispersion relation.
11.2 Acoustic waves with fnite amplitudes
We now have to study Eq.11.4 and 11.5, where we would like to replace the di±erentials in
ρ
by those in
c
s
. Using Eq.11.3
c
s
=
c
s,
0
ρ
ρ
0
!
(
γ

1)
/
2
⇒
dc
s
=
c
s
ρ
dρ
(11
.
10)
we obtain
∂
∂t
2
γ

1
c
s
!
+
u
∂
∂x
2
γ

1
c
s
!
+
c
s
∂u
∂x
= 0
(11
.
11)
∂u
∂t
+
u
∂u
∂x
+
c
s
∂
∂x
2
γ

1
c
s
!
= 0
(11
.
12)
Adding and subtracting these two coupled equations yields two individual equations
"
∂
∂t
+ (
u
+
c
s
)
∂
∂x
#
u
+
2
γ

1
c
s
!
= 0
(11
.
13)
"
∂
∂t
+ (
u

c
s
)
∂
∂x
#
u

2
γ

1
c
s
!
= 0
(11
.
14)
The solutions are
R
≡
u
+
2
γ

1
c
s
= const
.
on characteristic
dx
dt
=
u
+
c
s
(11
.
15)
L
≡
u

2
γ

1
c
s
= const
.
on characteristic
dx
dt
=
u

c
s
(11
.
16)
Let us assume there was on one wave moving to the right (in positive xdirection), so the
L
solution is a strict constant. The volume in front of the wave is therefore unperturbed and
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.
 Fall '08
 RABE
 Physics

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