Math 316 – Assignment 8
Due: Wednesday, March 23, 2011
1. Consider the wave equation utt = uxx for an infinite string (x in (−∞, ∞)), with initial
conditions
u(x, 0) = f (x),
ut (x, 0) = g(x).
a) Write out d’Alembert’s solution u(x, t).
b) Sketch the solution for t = 1 and t = 2 and t = 4, and explain its behavior as t increases when g(x) = 0 and f (x) is given by the following graph:
f (x)
6
1
@
@
@
-3
-2
-1
@
@
@
1
2
3
- x
4
c) Consider now f (x) = 0 and

 1 if − 6 < x < −2,
1 if 2 < x < 4,
g(x) =

0 otherwise.
Sketch the characteristic curves originating from x0 = −6, −2, 2, 4. Calculate the
solutions u(x, 1) and u(x, 2) and sketch their profiles.
Hint: Use the characteristic curves to distinguish several cases.
2. Find the solution to the following wave equation:
utt = c2 uxx ,
0 < x < 1, t > 0,
u(0, t) = 0, u(1, t) = 0,
u(x, 0) = 2 sin(2πx) + 3 sin(3πx), ut (x, 0) = sin(πx).
3. Find the solution to the following wave equation:
utt = c2 uxx ,
0 < x < L, t > 0,
ux (0, t) = 0, ux (L, t) = 0,
u(x, 0) = f (x), ut (x, 0) = g(x).