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An Introduction to Partial Differential Equations by Jacob Rubinstein, Yehuda Pinchover

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4

The one-dimensional wave equation

4.1 Introduction

In this chapter we study the one-dimensional wave equation on the real line. The canonical form of the wave equation will be used to show that the Cauchy problem is well-posed. Moreover, we shall derive simple explicit formulas for the solutions. We also discuss some important properties of the solutions of the wave equation which are typical for more general hyperbolic problems as well.

4.2 Canonical form and general solution

The homogeneous wave equation in one (spatial) dimension has the form

image

(4.1)

where c ∈ is called the wave speed, a terminology that will be justified in the discussion ...

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