A uniform asymptotic analysis of dispersive wave motion across a space-time shadow boundary

The one dimensional Klein-Gordon equation with spatially varying coefficients and with amplitude modulated high frequency signaling data is analyzed. A formal uniformly valid asymptotic expansion of the solution across a space-time shadow boundary is obtained with the help of two families of rays. These rays may also give rise to shadow regions. The asymptotic expansion involves three functions, a Fresnel function and two successive Bessel functions of integer order.