Wave solutions under complex space–time shifts

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Abstract
The idea of analytic continuation in both time and spatial source variables is adopted here to seek new space–time solutions of the wave equation. The solutions obtained depend not only on the complex space–time-shift parameters but on the specific temporal source behavior, which, in principle, can be selected at will. Three choices are investigated explicitly: The time-harmonic excitation leads to the well-known and widely applied time-harmonic Gaussian beams; the temporal impulse and the Gaussian excitations yield new solutions and are of particular interest, as they possess several desirable properties discussed in the text. A major application for the obtained solutions is their use as basis functions for generalized space–time-field representations. This leads to certain requirements. First, the solutions must form a complete basis. Second, they must be simple. Simplicity, here, implies the capability to calculate the expansion coefficients at an acceptable computational cost. This requirement is nontrivial, as the available solutions do not constitute an orthogonal basis. The field response to the Gaussian excitation is shown to be simple in this sense.

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