This paper presents analytical results on propagation of acoustic unipolar pulses in media with hysteretic nonlinearity characterized by a power law with an arbitrary exponent. Exact solutions for pulse profiles are derived for 1D propagation in the case of integer power-law exponents. For fractional ones, numeric and approximate analytical solutions are obtained. These results are used for interpretation of experimental data from the study by Y. Yasumoto, et al. [Acta Acustica united with Acustica 30(5) (1974) 260–267], where propagation of acoustic pulses in aluminum samples with different degree of annealing was observed. It is found that the dependence of the received pulse amplitude on the excitation amplitude corresponds to exponent n = 4 in the hysteretic equation of state, the nonlinear parameters of which increase with increasing temperature annealing.