Wave mixing equations describing the interaction of short-wavelength sound waves and entropy waves in two-fluid cosmic ray hydrodynamics in a non-uniform, large-scale, background flow in one Cartesian space dimension are investigated. The wave interaction coefficients depend on large-scale gradients in the background flow, and consist of two physically distinct components. The first component consists of wave-damping terms due to the diffusing cosmic rays, plus squeezing instability terms associated with the large-scale cosmic ray pressure gradient. These effects were first investigated by Drury and Dorfi in a study of the propagation of short-wavelength WKB sound waves in cosmic-ray-modified flows and shocks. The second component describes gas dynamical wave mixing effects due to gradients of the gas entropy S and the gas dynamical Riemann invariants (R±) of the background flow. A Green function solution is used to illustrate the coupling of the backward and forward sound waves for the case of a uniform background flow, in which the coupling coefficients depend on the parameter α = a2c/2κ, where ac is the cosmic-ray ‘sound speed’ and κ is the hydrodynamical cosmic-ray diffusion coefficient. Analytical WKB approximation methods and numerical simulations are used to investigate the modifications of the cosmic ray squeezing instability by wave mixing in cosmic-ray-modified shocks and pressure balance structures. Astrophysical applications to instabilities in supernova remnant shocks are discussed.