Thermoelasticity and generalized thermoelasticity viewed as wave hierarchies

It is seen how to write the standard form of the four partialdifferential equations in four unknowns of anisotropic thermoelasticityas a single equation in one variable, in terms of isothermaland isentropic wave operators. This equation, of diffusive type,is of the eighth order in the space derivatives and seventhorder in the time derivatives and so is parabolic in character.After having seen how to cast the 1D diffusion equation intoWhitham's wave hierarchy form, it is seen how to recast thefull equation, for unidirectional motion, in wave hierarchyform. The higher order waves are isothermal and the lower orderwaves are isentropic or purely diffusive. The wave hierarchyform is then used to derive the main features of the solutionof the initial-value problem, thereby bypassing the need foran asymptotic analysis of the integral form of the exact solution.The results are specialized to the isotropic case. The theoryof generalized thermoelasticity associates a relaxation timewith the heat flux vector and the resulting system of equationsis hyperbolic in character. It is also seen how to write thissystem in wave hierarchy form which is again used to derivethe main features of the solution of the initial-value problem.Simpler results are obtained in the isotropic case.