Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 39A / b

Each Fourier mode has zero initial velocity, soAlong $x=Vt$ this isPutThe two phases have stationary points at $-k_0$ and $k_0$, respectively. At those points,

Because the initial field is real, $a(-k)=a(k{)}^{\ast }$. Applying the stationary phase method to the two conjugate contributions gives the stationary-phase asymptotic of a Klein-Gordon wave along a subluminal ray:Equivalently, if $a(k_0)=|a(k_0)|e^{i\delta }$,

If the real initial displacement is even, then $a(k)$ is real and even. Assume $a(k_0)\ne 0$ and setWhen $a(k_0)>0$, the leading result from part (i) is a positive multiple ofIts upward zero crossings of an oscillatory stationary-phase tail occur when the phase is $3\pi /2$ modulo $2\pi$. ThereforeIf $a(k_0)<0$, the sign is reversed and the upward crossings instead satisfyThe slow derivative of the $t^{-1/2}$ envelope changes these times only beyond leading order.