Stationary-phase asymptotic of a Klein-Gordon wave along a subluminal ray

For zero initial velocity and Fourier amplitude $a(k)$, observation along $x=Vt$ with $0\le V<c$ selects the two stationary wavenumbers $\pm k_0$, whereWriting $s=\sqrt{c^2-V^2}$, the leading oscillation has angular frequency $As$ and amplitude proportional to $t^{-1/2}$.

If the leading asymptotic is $C{t}^{-1/2}\cos (\Omega t+\gamma )$ with $C>0$, its upward zero crossings satisfyto leading order.