General Theory of Acoustic Wave Propagation in Liquids and Gases

We study the propagation of small-amplitude acoustic waves in liquids and gases and use the hydrodynamic equations to obtain an exact dispersion equation. This equation in dimensionless variables contains only two material constants $p$ and $q$. We solve the dispersion equation, obtaining an exact solution that holds for all values of the parameters and all frequencies up to hypersonic, and thus analytically establish exactly how the speed of sound $c$, the wave vector $k$, and the damping factor $x$ depend on the frequency $\omega$ and the dimensionless material constants $p$ and $q$. Studying the behavior of the solution in the sonic and ultrasonic frequency bands for $\omega<10^7$ с$^{-1}$ results in an expression for the damping factor, which differs from the Kirchhoff formula. The speed of sound $c$ and the wave vector $k$ are shown to have finite nonzero values for all hypersonic frequencies. At the same time, there exists a certain maximum frequency value, $\omega_{\max}\approx10^{11}$–$10^{12}$ с$^{-1}$, at which the damping factor $x$ is zero. This frequency determines the boundary of the applicability domain for the hydrodynamic equations.