The jumps in Helmholtz free energy and stress across a shock wave front are obtained by using the Taylor's series expansion of the corresponding functions of arbitrarily constrained thermoelastic solids. The linear terms in the expressions of jumps are considered alone to account for linear weak shock waves. The generalized thermomechanical constraint functions and purely mechanical constraint functions are treated separately. in the case of the thermoelastic solids with purely mechanical constraints, the propagation condition of weak shock waves is found to be similar to the propagation condition of homothermal acceleration waves. In the case of the thermoelastic solids with general thermomechanical constraints, the corresponding propagation condition is found to be similar to the propagation condition of homentropic acceleration waves. The speed of propagation of weak shock waves is obtained in a linear thermoelastic solid subject to the purely mechanical constraint of inextensibility.