ACM 95/100c
May 14, 2004
Problem Set VI
0. (2 pts) Write down your gradingsection number.
1. (10 pts) Sketch the solutions to the wave equation:
u
(
x, t
) =
1
2
[
u
(
x
+
ct,
0) +
u
(
x

ct,
0)] +
1
2
c
x
+
ct
x

ct
u
t
(
τ,
0)
dτ
for various values of
t
arising from the following initial conditions:
(a) (5 pts)
u
(
x,
0) = 0
,
u
t
(
x,
0) = sin
ωx
(
ω
is a constant), and
(b) (5 pts)
u
(
x,
0) = 0
,
u
t
(
x,
0) =
1
for
0
< x <
1

1
for

1
< x <
0
.
0
for

x

>
1
Note that in part (b) the velocity
u
t
(
x,
0) is discontinuous; the appropriate interpretation of the
problem is that we consider a sequence of initial conditions, described by continuous (smooth)
functions that approach the given initial condition in the limit.
In this sense it is possible to
give meaning to discontinuous initial conditions. Note that in contrast to the heat equation which
smoothes out discontinuities, solutions to the wave equation with discontinuous initial conditions
retain their discontinuity for
t >
0.
2. (15 pts)
(a) (5 pts) Consider the solution
u
(
x, t
) of the wave equation
u
tt
=
c
2
u
xx
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 Spring '09
 NilesA.Pierce
 Energy, Kinetic Energy, Boundary value problem, Partial differential equation, wave equation

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