Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 20 / c

For the two travelling profiles, the initial conditions implySince $v\ne 0$, $g-f$ is constant. One may therefore choosewhere the constants cancel in the sum. The D'Alembert formula then givesThis agrees exactly with the restricted-domain solution from part (b). The whole-line cosine profile already has zero spatial derivative at both $x=0$ and $x=\pi$ for every time, so its restriction satisfies the Neumann boundary conditions without further modification.

The standing wave oscillates about the constant level $1/2$, with time period $2\pi /|v|$. Suitable snapshots areThe nonconstant amplitude shrinks to zero at a quarter-period, reverses sign at half a period, and returns after a full period. The nodes of a standing wave of the oscillatory part, $x=\pi /2+k\pi$, remain at height $1/2$.