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Chapter 5: Waves
21
The equation for a harmonic traveling plane wave is:
u
(
±
x, t
)=ˆ
u
cos(
±
k
·
±
x
±
ω
t
+
ϕ
)
If waves reFect at the end of a spring this will result in a change in phase. A ±xed end gives a phase change of
π
/
2
to the reFected wave, with boundary condition
u
(
l
)=0
. A lose end gives no change in the phase of the
reFected wave, with boundary condition
(
∂
u/
∂
x
)
l
=0
.
If an observer is moving w.r.t. the wave with a velocity
v
obs
, he will observe a change in frequency: the
Doppler effect
. This is given by:
f
f
0
=
v
f

v
obs
v
f
.
5.2.2
Spherical waves
When the situation is spherical symmetric, the homogeneous wave equation is given by:
1
v
2
∂
2
(
ru
)
∂
t
2

∂
2
(
ru
)
∂
r
2
=0
with general solution:
u
(
r, t
)=
C
1
f
(
r

vt
)
r
+
C
2
g
(
r
+
vt
)
r
5.2.3
Cylindrical waves
When the situation has a cylindrical symmetry, the homogeneous wave equation becomes:
1
v
2
∂
2
u
∂
t
2

1
r
∂
∂
r
±
r
∂
u
∂
r
²
=0
This is a Bessel equation, with solutions which can be written as Hankel functions. ²or suf±cient large values
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 Spring '10
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