The relationship between the ultrasonic nonlinearity parameter for solids and the acoustic Grüneisen number has been derived for longitudinal ultrasonic wave propagation in the pure mode directions of cubic crystals and isotropic solids. Agreement between the acoustic Grüneisen number and the thermodynamic Grüneisen parameter is best for ultrasonic harmonic generation measurements along the [100] direction of a cubic crystal. Comparison of the temperature curves of the acoustic Grüneisen number of copper shows that the acoustic Grüneisen number generally follows the temperature dependence of the lattice contribution to the thermodynamic Grüneisen parameter.