Sound Attenuation and Dispersion near the Liquid-Gas Critical Point
Abstract
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 $\ensuremath{\epsilon}$, the dimensionless temperature distance from the critical point. The attenuation behaves as ${f}^{2}{\ensuremath{\epsilon}}^{\ensuremath{-}3\ensuremath{\nu}\ensuremath{-}\frac{\ensuremath{\alpha}}{2}}$ for $0\ensuremath{\le}f\ensuremath{\ll}{\ensuremath{\epsilon}}^{3\ensuremath{\nu}}$, and as ${f}^{2\ensuremath{-}\frac{2p}{3}}{\ensuremath{\epsilon}}^{\ensuremath{-}\frac{3\ensuremath{\alpha}}{2}}$ for ${\ensuremath{\epsilon}}^{3\ensuremath{\nu}}\ensuremath{\ll}f\ensuremath{\ll}{\ensuremath{\epsilon}}^{\ensuremath{\nu}}$, where $p$ is the exponent which appears in the wave-number ($\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$)-dependent correlation of the order parameter expressed as ${A}_{1}{k}^{\ensuremath{-}2+\ensuremath{\eta}}+{A}_{2}{\ensuremath{\epsilon}}^{1\ensuremath{-}\ensuremath{\alpha}}{k}^{\ensuremath{-}2+\ensuremath{\eta}\ensuremath{-}\ensuremath{\rho}}$, when $k$ is much greater than the inverse correlation range of critical fluctuations. The relative sound-velocity change with $f$ behaves as ${f}^{\frac{3}{2}}{\ensuremath{\epsilon}}^{\ensuremath{-}\frac{9\ensuremath{\nu}}{2}}$ for $0\ensuremath{\le}f\ensuremath{\ll}{\ensuremath{\epsilon}}^{3\ensuremath{\nu}}$, as ${f}^{1\ensuremath{-}\frac{2p}{3}}{\ensuremath{\epsilon}}^{\ensuremath{-}\ensuremath{\alpha}}$ if $p\ensuremath{\le}\frac{3}{2}$, and as ${f}^{0}{\ensuremath{\epsilon}}^{0}$ if $p>\frac{3}{2}$ for ${\ensuremath{\epsilon}}^{3\ensuremath{\nu}}\ensuremath{\ll}f\ensuremath{\ll}{\ensuremath{\epsilon}}^{\ensuremath{\nu}+\frac{\ensuremath{\alpha}}{2}}$. The explicit expressions for the attenuation and dispersion are given for $f\ensuremath{\sim}{\ensuremath{\epsilon}}^{3\ensuremath{\nu}}$.