PHYSICS 218
SOLUTION TO HANDOUT 6
Created: September 25, 2004
8:55pm
Last updated: September 26, 2004
8:14am
Gaussian Wave Packet
Consider a transverse wave propagating along an infinite onedimensional string which satisfies the
wave equation
∂η
∂x
2

1
c
2
∂η
∂t
2
,
where
η
(
x, t
) is the displacement from the string’s equilibrium position.
The initial profile of the
wave is given by
η
(
x, t
= 0) =
A
e

x
2
,
where
A
is a constant independent of time. Waves of this form are called Gaussian. Let the initial
velocity profile be given by
∂
∂t
η
(
x, t
= 0) =
A c x
e

x
2
,
where
c
is the wave speed.
1. Let the uniform in the string be
τ
0
, what is the linear mass density
λ
0
of the string?
Solution:
This is to refresh your memory on wave equation for onedimensional string. Recall
that
c
=
τ
0
/λ
0
. So
λ
0
=
τ
0
/c
2
.
2. Let’s assume that the wave can be expressed in terms of two travelling waves, namely
η
(
x, t
) =
f
1
(
x

c t
) +
f
2
(
x
+
c t
)
.
Let the righttravelling wave
f
1
have boundary condition
f
1
(
x
= 0) =
3
4
A
. Find the equation
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 Fall '04
 POLLACK
 Physics, Magnetism, wave equation, Gaussian wave packet, wave proﬁle

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