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

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 $2c$.

The multivariable chain rule gives ${\partial }_x={\partial }_{\xi }+{\partial }_{\eta }$ and ${\partial }_t=\alpha {\partial }_{\xi }+\beta {\partial }_{\eta }$. Substitution into the differential equation yieldsThe coordinate transformation is invertible exactly when $\alpha \ne \beta$.

Eliminate the two pure second derivatives by taking $\alpha ,\beta$ to be the roots of $z^2+2Bz+C=0$. They are distinct real numbers exactly whenThus the required function is $H(B)=B^2$. The mixed coefficient becomes $4(C-B^2)\ne 0$, so the transformed equation is $y_{\xi \eta }=0$. Integrating once in each coordinate giveswith arbitrary sufficiently smooth functions $F,G$. This is the characteristic form of a strictly hyperbolic partial differential equation.