Asymptotic behavior relative to a large parameter of the solution of the Fok–Klein–Gordon equation in the case of a discontinuous initial condition

One considers the problem of the asymptotic behavior for $k\to+\infty$ of the solution of the Cauchy problem:
$$ u_{tt}-u_{xx}+k^2u=0;\qquad u\mid_{t=0}=\theta(x),\quad u_t\mid_{t=0}=0\ (t>0\text{ -- fixed}). $$
Here $\theta(x)$ is the Heaviside function. In the neighborhood of the characteristics $x=\pm t$ function $u(x,t)$ oscillates exceptionally fast (the wavelength is of order $k^{-2}$). Near the $t$ axis the asymptotics of $u(x,t)$ contains the Fresnel integral.