Incompressibility at uniform temperature or entropy in isotropic thermoelasticity

It is known that two stable longitudinal waves may propagate in each direction through an isotropic thermoelastic material. It is shown that two longitudinal waves may propagate in each direction through an isotropic thermoelastic material which is incompressible at uniform temperature, one wave being stable and the other unstable. As frequency varies these modes occupy parts of a rectangular hyperbola in the complex plane of squared wave speeds. It is further shown that only one longitudinal wave may propagate if the isotropic thermoelastic material is incompressible at uniform entropy rather than temperature. This is stable wave. The corresponding mode occupies a semicircle in the complex plane introduced above.