Astro 405/505, fall semester 2005
Homework, 4th set, return before Friday, November 4, 4pm.
Don’t forget to give your name.
Problem 7: Fourier transformation
In class we derived the general solution of the lowamplitude acoustic wave equation as (11.9)
∂
2
ρ
1
∂t
2
=
c
2
s,
0
∂
2
ρ
1
∂x
2
⇒
ρ
1
=
f
(
x

c
s,
0
t
) +
g
(
x
+
c
s,
0
t
)
where
f
and
g
are arbitrary functions that can be written as the superposition of waves with
amplitude spectrum
˜
f
f
(
y
) =
1
√
2
π
Z
∞
∞
dk
˜
f
(
k
) exp(
ı k y
)
˜
f
(
k
) =
1
√
2
π
Z
∞
∞
dy f
(
y
) exp(

ı k y
)
The integral transformation that links
f
and
˜
f
is called Fourier transformation.
Solve the lowamplitude acoustic wave equation for a sound speed that depends on the fre
quency
ω
or wavenumber
k
. Can I still write the solution as a Fourier integral depending on
y
=
x
±
c
s
t
?
The wave packets
f
and
g
now consist of waves that propagate with di±erent phase velocities
ω/k
.
Assume a wave packet that contains only waves with frequencies close to
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 Fall '08
 RABE
 Physics, Atom, Work, Fundamental physics concepts, Energy density, acoustic wave equation, lowamplitude acoustic wave

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