How Can Variable Changes Simplify the One-Dimensional Wave Equation?

[javicg]

I'm having trouble with the following related questions. Any help is appreciated.

(a) Show that a change of variables of the form $$\xi = px + qt$$, $$\eta = rx + st$$ can be used to reduce the one dimensional wave equation $$\frac1{c^2} u_{tt} = u_{xx}$$ to an equation of the form $$\frac{\partial^2 U}{\partial\xi \partial\eta} = 0$$. Hence show that the general solution of the wave equation is of the form $$u(x,t) = F(x + ct) + G(x - ct)$$, where F,G are arbitrary twice differentiable functions.

(b) Show that the solution of the wave equation for the infinite domain $$-\infty < x < \infty$$ subject to $$u(x,0) = f(x)$$ and $$u_t(x,0) = g(x)$$ can be written as $$u(x,t) = \frac12 [f(x + ct) + f(x - ct)] + \frac1{2c} \int_{x - ct}^{x + ct} g(y) dy.$$

This is called the D'Alembert solution.

 

[HallsofIvy]

What have you done on this yourself? It is a pretty direct, though tedious, exercise in using the chain rule to change variables in a differential equation.

$$\frac{\partial U}{\partial x}= \frac{\partial \xi}{\partial x}\frac{\partial U}{\partial \xi}+ \frac{\partial \eta}{\partial x}\frac{\partial U}{\partial \eta}$$