Exercise 49

Assume that all the given functions have continuous second-order partial derivatives. Show that any function of the form \(z = f(x + at) + g(x - at)\) is a solution of the wave equation \[ \frac{\partial^2 z}{\partial t^2} = a^2 \frac{\partial^2 z}{\partial x^2} \] [Hint: Let \(u = x + at, v = x - at\).]