Using the Walsh‐Christian formulation for the isentropes of a solid having the property of maximum free‐surface velocity, we obtain the first three terms in the expansion of the Gru¨neisen parameter in terms of the compression. In addition, we determine the coefficients of the quadratic and cubic terms in the expansion of the shock velocity in terms of the particle velocity. Finally, we show that the Gru¨neisen parameter for a maximum free‐surface velocity solid furnishes an upper bound to the Gru¨neisen parameter for a given solid. Calculations are given for aluminum and copper by way of illustration.