A compressible Maxwell-Cahn-Hilliard binary fluid mixture in conjunction with the Sanchez-Lacombe equation of state is used to develop a one-dimensional linear model that describes acousto-spinodal decomposition by pressure-induced phase separation in compressible polymer solutions in the absence of externally imposed flow. The integrated model extends previous work acousto-spinodal decomposition of compressible polymer solutions using Euler and Newtonian model fluids [1,2]. Acousto-spinodal decomposition by pressure-induced phase separation couples density wave phenomena with mass transfer driven by the spinodal decomposition process. For viscoelastic acousto-spinodal decomposition the relevant parameters are the Mach number Ma (ratio of diffusion speed to density wave speed) and the Reynolds Re and Deborah De numbers. Concentration gradients act as a sound source in the density wave equation that feeds back into the mass diffusion equation producing, under sufficiently low Ma oscillatory spinodal decomposition. This oscillatory effect is generally suppressed by viscosity and enhanced by elasticity. The limiting standing waves (1/Re -> 0, 1/Ma(2) -> 0, De -> 0) that can be generated include elasto-acoustic, viscoelastic, and visco-acoustic. Concentration fluctuations accompanied with density fluctuations generate velocity, which are a unique feature of acousto-spinodal decomposition. The role of elasticity on the rate of demixing and selection of length scales is characterized; it is found that at high viscosity and low mass diffusion, higher elasticity De is required to reach absolute maximum growth rate R-infinity(max), which corresponds to an incompressible Euler fluid. Total structure factor calculations are shown to be qualitatively consistent with experiments at low wave vectors. Acousto-spinodal decomposition is a novel process that offers new processing and characterization routes for polymer solutions and melts. (C) 2014 Elsevier B.V. All rights reserved.