A theorem on isotropic null vectors and its application to thermoelasticity

Abstract

Let A be a symmetric n x n matrix over an arbitrary field. We show that if A possesses a non-zero null vector x which is also isotropic (i. e. such that x∙ x = 0), then both the trace of the adjugate and the determinant of A vanish. For the complex field the converse is also true. In purely mechanical elasticity it is known that an isotropic wave amplitude vector (which corresponds to a circularly polarized wave) is necessarily associated with a double root of the characteristic equation of wave propagation (whose roots give essentially the squared wave speeds). We apply the preceding results to obtain concise conditions for the occurrence both of isotropic wave amplitudes and of double roots in the theory of thermoelastic wave propagation. In sharp contrast with the purely mechanical theory, we find that these two phenomena generally occur separately in thermoelasticity.