 Solution

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.