Multiple–scales perturbation methods are used to derive equations describing wave–wave interactions in two-fluid cosmic-ray hydrodynamics in one Cartesian space dimension. Two problems are considered: (a) the interaction between short-wavelength waves in a non-uniform large-scale background flow, and (b) wave interactions between long-wavelength waves propagating through a uniform background medium. The short-wavelength wave equations describe the interaction between the backward and forward thermal gas sound waves and the contact discontinuity eigenmode via ‘wave mixing’, in which the waves are reflected by gradients in the large scale background flow. The equations also contain quadratic wave interaction terms describing (i) the Burgers self-wave interaction term, (ii) mean–wave-field interaction terms in which a specific wave interacts with the mean field of the other waves, and (iii) three-wave resonant interactions that describe how a sound wave resonantly interacts with the contact discontinuity to generate a reverse sound wave. In the limit of no interaction between the cosmic rays and thermal gas, and for a uniform background state, the equations reduce to two coupled, integro-differential Burgers equations derived by Majda and Rosales for resonantly interacting waves in adiabatic gasdynamics. The short-wavelength equations also contain squeezing instability terms associated with the large-scale cosmic-ray pressure gradient, which were first investigated by Drury and Dorfi. A generalized wave-action equation, or canonical wave-energy equation, variational principles, and WKB analyses of the linearized equations are used to investigate the modification of the cosmic-ray squeezing instability by wave mixing. Coupled Burgers equations are also derived in the long-wavelength regime that describe resonant wave interactions for weak, diffusively smoothed cosmic-ray-modified shocks. In the latter equations, the cosmic-ray-modified sound waves can resonantly interact with either the contact discontinuity or the cosmic-ray pressure-balance mode to generate a reverse sound wave.
