The Relativistic Hermite Polynomials and the Wave Equation

Abstract

Abstract-Solutions of the homogeneous 2D scalar wave equation of a type reminiscent of the "splash pulse" waveform are investigated in some detail. In particular, it is shown that the "higher-order" solutions relative to a given "fundamental" one, from which they are obtained through a definite "generation scheme", come to involve the relativistic Hermite polynomials. This parallels the results of a previous work, where solutions of the 3D wave equation involving the relativistic Laguerre polynomials have been suggested. Then, exploiting a well known rule, the obtained wave functions are used to construct further solutions of the 3D wave equation. The link of the resulting wave functions with those analyzed in the previous work is clarified, the pertinent generation scheme being indeed inferred. Finally, solutions of the Klein-Gordon equation which relate to such Lorentzian-like solutions of the scalar wave equation are deduced.

An Adaptive Cavity Setup for Accurate Measurements of Complex Dielectric Permittivity

“…Let us consider in detail the TE case. The differential eigenvalue problem (1) is solved by a spectral method based on orthogonal polynomials, [21,22]. The eigenfunctions V e,d (z) are C ∞ everywhere apart from the interface, where they are only C 1 (C 0 in the TM case).…”