Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 6 / Solution

Set $\xi =x-ct$. The chain rule gives $u_t=-c{f}^{\prime }(\xi )$ and $u_x=f^{\prime }(\xi )$, so $u_t+c{u}_x=0$ for any differentiable $f$. The initial profile fixes $f(\xi )=\cos \xi$, givingThis is the translated initial profile, as described by the transport equation.