Random waves in one‐dimensional nonlinear elastic materials are considered. An integrodifferential equation is derived that governs the variation of the strain spectrum during propagation. By discretizing the spectrum with respect to the frequency, the spectral equation reduces to a system of ordinary differential equations for the discrete values of the spectrum. Numerical results are given for plane random longitudinal waves in an aluminum alloy. It is found that harmonic sound waves are generated for a narrow‐band spectrum, the power spectrum grows faster for higher frequencies for a wideband spectrum, and secondary waves are generated due to the interaction of two bands of spectra.