The governing mechanical and thermal equations in Lagrangian co-ordinates are briefly derived. The results of the isentropic theory are reviewed; a difficulty in discontinuity analysis is stated but not completely resolved. Constant profile solutions of the full thermomechanical equations for conducting solids are discussed and the dilatational profile treated in detail. Strong discontinuities may arise and it is shown that the temperature must be continuous. Applications to real materials are discussed and further problems are suggested.