= Solution
The <standing wave> oscillates about the constant level $1/2$, with time period $2\pi/|v|$. Suitable snapshots are
$$
\begin{aligned}
t=0:&\quad\psi=\tfrac12(1+\cos x),\\
t=\frac{\pi}{3|v|}:&\quad\psi=\tfrac12+\tfrac14\cos x,\\
t=\frac{\pi}{2|v|}:&\quad\psi=\tfrac12,\\
t=\frac{\pi}{|v|}:&\quad\psi=\tfrac12(1-\cos x),\\
t=\frac{2\pi}{|v|}:&\quad\psi=\tfrac12(1+\cos x).
\end{aligned}
$$
The 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 $1/2$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2-q20-standing-wave.png]
{title=Time evolution of the whole-line Neumann-compatible standing wave}
{description=Snapshots at phases $|v|t=0$, $\pi/3$, $\pi/2$ and $\pi$ illustrate contraction, flattening and reversal of the cosine profile. A full-period snapshot repeats the initial curve.}
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