Sound Attenuation and Dispersion near the Liquid-Gas Critical Point

The attenuation and dispersion of sound near the gas-liquid critical point are studied theoretically using the author's extended mode-mode coupling theory. The results differ in the different regions of the sound-wave frequency $f$ expressed in a dimensionless unit and of $\epsilon$, the dimensionless temperature distance from the critical point. The attenuation behaves as $f^2{\epsilon }^{-3\mathrm{}\nu -\frac{\alpha }{2}}$ for $0\le f\ll {\epsilon }^{3\nu }$, and as $f^{2-\frac{2p}{3}}{\epsilon }^{-\frac{3\alpha }{2}}$ for ${\epsilon }^{3\nu }\ll f\ll {\epsilon }^{\nu }$, where $p$ is the exponent which appears in the wave-number ($\overrightarrow{\mathrm{k}}$)-dependent correlation of the order parameter expressed as $A_1{k}^{-2+\eta }+A_2{\epsilon }^{1-\alpha }{k}^{-2+\eta -\rho }$, when $k$ is much greater than the inverse correlation range of critical fluctuations. The relative sound-velocity change with $f$ behaves as $f^{\frac{3}{2}}{\epsilon }^{-\frac{9\nu }{2}}$ for $0\le f\ll {\epsilon }^{3\nu }$, as $f^{1-\frac{2p}{3}}{\epsilon }^{-\alpha }$ if $p\le \frac{3}{2}$, and as $f^0{\epsilon }^0$ if $p>\mathrm{}\frac{3}{2}$ for ${\epsilon }^{3\nu }\ll f\ll {\epsilon }^{\nu +\frac{\alpha }{2}}$. The explicit expressions for the attenuation and dispersion are given for $f\sim {\epsilon }^{3\nu }$.