Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics V / 5 / a

For $c>0$ and $L>0$, D'Alembert's formula for the wave equation on the full real line isFor the specified initial data, define the cumulative initial-velocity integralThen $\boxed{y(x,t)=[A(x+ct)-A(x-ct)]/(2c)}$. The initial velocity jumps at $\pm L$, so the resulting solution is interpreted piecewise classically, or as a weak wave-equation solution across the corresponding characteristic curves.

At $ct=L/2$, the interval of integration has half-width $L/2$. Evaluating the cumulative integral in part a gives the even functionIt has value $L^2/(8c)$ at zero and maxima $L^2/(4c)$ at $x=\pm L/2$, then falls to zero at the two wavefronts.

At $x=L/2$, the integration interval first moves through positive arguments, then crosses zero, and finally contains all of $[-L,L]$. The result isThe final plateau is the integral of the entire initial velocity profile divided by $2c$.