Exact Solution for the Propagation of Electromagnetic Pulses Over a Highly Conducting Spherical Earth
An analytic solution for the propagation of electromagnetic pulses by ground waves, a problem previously treated by the saddle-point method or numerical integration. In this study, the various integrals are evaluated exactly. First, the problem is formulated and the limitations of the saddle-point method are discussed. The required integrals are then evaluated exactly as products of Airy functions. Detailed results are worked out for a step and ramp source; the technique is also applied to a waveform of nuclear origin that was previously investigated by numerical integration. The step and ramp function responses are shown to be well approximated by a single mode for early and intermediate times. The saddle-point approximation is valid only for times before the peak, but is a good approximation out to 50 times that duration. At late times, the single-mode exact response goes through zero, reaches a broad negative maximum, and then returns slowly to zero, while the saddle-point single mode and the mode sum responses remain positive; the ramp function displays corresponding behavior. For the nuclear waveform, the exact curves have the same general character and about the same peak values as the numerical integration results, but the exact curves consistently display a faster time scale, and the spectrum peaks at a higher frequency.