Model equations with cubic nonlinearity are developed for a plane shear wave of finite amplitude in a relaxing medium. The evolution equation for progressive waves is solved analytically for a jump in stress that propagates into an undisturbed medium. Weak-shock theory is used to determine the amplitude and location of the shock when the solution predicts a multivalued waveform. The solution is similar to that obtained by Polyakova, Soluyan, and Khokhlov [Sov. Phys. Acoust. 8, 78-82 (1962)] for a compressional wave with quadratic nonlinearity in a relaxing fluid. Numerical simulations illustrate the effect of relaxation on shock formation in an initially sinusoidal shear wave. The minimum source amplitude required for an initially sinusoidal waveform to develop shocks in a relaxing medium is determined as a function of the dispersion and relaxation time. Limiting forms of the evolution equation are considered, and analytical solutions incorporating weak-shock theory are presented in the high-frequency limit. A Duffing-type model for a nonlinear shear-wave resonator is developed and investigated.
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