Plane sound waves of low amplitude in a gas-dust environment with polydispersed particles

The problem of the propagation of plane sound waves of small amplitudes in a mixture of an isothermal carrier gas and solid particles of various sizes is formulated on the basis of a multi-liquid macroscopic model of the medium. In the model, the dispersed phase is considered as $N$ fractions of monodisperse particles, and the dynamics of each fraction is described using the equations of a continuous medium in which does internal pressure is absent. The fractions exchange momenta with the carrier gas, but not with each other. The whole mixture is acted upon by the total pressure determined by the motion of gas molecules, and the dust particles are considered buoyant. An analytical solution of the problem is obtained using the Fourier method and analysis of variance. In the general case for an arbitrary value of the relaxation time, the solution is found numerically using the developed and published code. In special cases (infinitely small time of velocity relaxation or relaxation equilibrium and infinitely long time of velocity relaxation or frozen equilibrium), the effective velocity of sound in the gas-dust medium is determined and used to obtain simple analytical representations of the solution of the problem.