Stationary-phase asymptotic of a Klein-Gordon wave along a subluminal ray
= 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 $0\leq V<c$ selects the two stationary wavenumbers $\pm k_0$, where
$$
k_0=\frac{AV}{\sqrt{c^2-V^2}}.
$$
Writing $s=\sqrt{c^2-V^2}$, the leading oscillation has angular frequency $As$ and amplitude proportional to $t^{-1/2}$.