It is seen how to write the standard form of the four partial differential equations in four unknowns of anisotropic thermoelasticity as a single equation in one variable, in terms of isothermal and isentropic wave operators. This equation, of diffusive type, is of the eighth order in the space derivatives and seventh order in the time derivatives and so is parabolic in character. After having seen how to cast the 1D diffusion equation into Whitham's wave hierarchy form, it is seen how to recast the full equation, for unidirectional motion, in wave hierarchy form. The higher order waves are isothermal and the lower order waves are isentropic or purely diffusive. The wave hierarchy form is then used to derive the main features of the solution of the initial-value problem, thereby bypassing the need for an asymptotic analysis of the integral form of the exact solution. The results are specialized to the isotropic case. The theory of generalized thermoelasticity associates a relaxation time with the heat flux vector and the resulting system of equations is hyperbolic in character. It is also seen how to write this system in wave hierarchy form which is again used to derive the main features of the solution of the initial-value problem. Simpler results are obtained in the isotropic case.