Causal Theories of Evolution and Wave Propagation in Mathematical Physics

Abstract

There are still many phenomena, especially in continuum physics, that are described by means of parabolic partial differential equations whose solution are not compatible with the causality principle. Compatibility with this principle is required also by the theory of relativity. A general form of hyperbolic operators for the most frequently occurring linear governing equations in mathematical physics is written down. It is then easy to convert any given parabolic equation to the hyperbolic form without necessarily entering into the cause of the inadequacy of the governing equation. The method is verified on the well-known example of Timoshenko’s correction of the Bernoulli–Euler–Rayleigh beam equation for flexural motion. The “Love–Rayleigh” fourth-order differential equations for the longitudinal and torsional wave propagation in the rod is generalized with this method. The hyperbolic version (not to mention others) of the linear Korteweg–de Vries equation and of the “telegraph” equation governing electromagnetic wave propagation through relaxing material are given. Lagrangians of all the equations studied are listed. For all the reasons given we believe the hyperbolic governing equations to be physically and mathematically more realistic and adequate.