Beginning Partial Differential Equations, 3rd Edition

Chapter 3

Solutions of the Wave Equation

This chapter deals with models of wave motion under a variety of conditions, including fixed ends, moving ends, forcing terms, and damping effects. We also delve into the structure of solutions of the wave equation, continuous dependence of solutions on initial and boundary data, and solutions in unbounded media and in higher dimensions.

3.1 Solutions on Bounded Intervals

3.1.1 Fixed Ends

Consider a homogeneous string or wire of length L, fixed at its ends. At time zero it is displaced and released with a given initial velocity. The initial-boundary-value problem for the function describing the wave motion is:

(3.1)

Separate the variables in problem 3.1. Put u(x, t) = X(x)T(t) into the wave equation to get

so

Both sides must be constant, because x and t are independent and either could be fixed while the other varies in this equation. Therefore, for some λ to be determined,

Then

Reasoning as we did with the heat equation, the boundary condition ...