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Chapter 8

Initial Boundary Value Problem for

the Wave Equation

8.1 Solved Problems

Exercise 159. Consider a string of length L with a fixed left end a free right end.
Initially the string is at rest with displacement f(x). Find the motion of the string
by solving,

2
u = c u , 0 < x < L, t > 0,
tt
xx
u(0, t) = 0, u (L, t) = 0,
x
u(x, 0) = f(x), u (x, 0) = 0,
t
with separation of variables.

Solution: Use separation of variables to find eigen-solutions of the partial differential
equation that satisfy the homogeneous boundary conditions. There will be two
eigen-solutions for each eigenvalue. Expand u(x, t) in a series of the eigen-solutions.
Use the two initial conditions to determine the constants.

2
u = c u , 0 < x < L, t > 0,
xx
tt
u(0, t) = 0, u (L, t) = 0,
x
u(x, 0) = f(x), u (x, 0) = 0,
t
We substitute the separation of variables u(x, t) = X(x)T(t) into the partial dif-
ferential equation.

2
(XT) = c (XT) xx
tt
T 00 X 00
= = −λ 2
2
c T X
With the boundary conditions at x = 0, L, we have the ordinary differential equa-
tions,

00
2 2
T = −c λ T,

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