Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 19 / a / Solution

Assume $c\ne 0$, so $(\xi ,\eta )=(x-ct,x+ct)$ is an invertible coordinate change. The chain rule givesFor a twice continuously differentiable solution the mixed partial derivatives commute. Squaring the two operators therefore givesThus the one-dimensional wave equation becomes