Damped Wave Conduction and Relaxation in Cylindrical and Spherical Coordinates

Abstract

The damped wave conduction and relaxation equation was solved for infinite, cylindrical, and spherical mediums using the method of relativistic transformation of variables. The transformation η = r2 - X2 was used to transform the hyperbolic partial differential equation (which describes the wave temperature) into a Bessel differential equation in the transformation variable only. The solution is characterized by three distinct regimes. For interior points in the medium, there is a thermal inertial or zero transfer regime. The solution for the cylinder was characterized by a Bessel composite function in space and time of the half-order and first kind in the open interval X > T and by a modified composite function in space and time of the half-order and first kind in the open interval T > X. For the sphere, the solution was characterized by a Bessel composite function in space and time of the first kind and first order in the open interval X > τ and by a modified composite function in space and time of the first order and first kind in the open interval T > X. The regimes are illustrated in figures and the solution is free of singularities.