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Chapter 8

Initial Boundary Value Problem

for the Wave Equation

8.1 Introduction

Fourier’s method for solving linear PDEs is based on the technique of separation of vari-
ables. Let us outline the main steps of this technique. First we search for solutions of
the homogeneous PDE that are called product solutions (or separated solutions). These
solutions have the special form

u(x, t) = X(x)T(t),

and in general they should satisfy certain additional conditions. In many cases, these
additional conditions are just homogeneous boundary conditions. It turns out that X and
T should be solutions of linear ODEs that are easily derived from the given PDE. In the
second step, we use a generalization of the superposition principle to generate out of the
separated solutions a more general solution of the PDE, in the form of an infinite series
of product solutions. In the last step we compute the coefficients of this series.
Since the separation of variables method relies on several deep ideas and also involves
several technical steps, we present in the current chapter the technique for solving several
relatively simple problems without much theoretical justification. We note that Fourier’s
method is based on constructing solutions of a specific type.

8.2 Dirichlet conditions

Imagine we have a tensioned guitar string of length l. Suppose we only consider vibrations
in one direction. That is, let x denote the position along the string, let t denote time,
and let u(x, t) denote the displacement of the string from the rest position.
Assume that the ends of the string are fixed in place:
u(0, t) = 0 and u(l, t) = 0.

Note that we have two conditions along the x− axis as there are two derivatives in this
direction.

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