The linear weak shock wave (acoustic wave) propagation and the existence of shear bands are examined in finitely deformed thermoelastic solids within the framework of the theory of singular surfaces. The jumps of certain field variables across the shock wave front are obtained by using Taylor series expansions of them. The propagation condition of shock waves in a thermoelastic solid is obtained by using the strain-energy function corresponding to Duhamel-Neumann expression. The propagation speeds of weak shock waves are determined for a particular state of deformation, that is, general dilation. The formation of shear bands and the magnitudes of critical stretches are obtained for the deformation states of uniaxial, biaxial extension and for uniform dilation.