Chapter 12.4, Problem 79E

Wave equation Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are governed by the one-dimensional wave equation

     $\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2},$

where u(x, t) is the height or displacement of the wave surface at position x and time t, and c is the constant speed of the wave. Show that the following functions are solutions of the wave equation.

79.    u(x, t) = A f(x + ct) + B g(x – ct), where A and B are constants and f and g are twice differentiable functions of one variable