Velocity of Sound, Density, and Grüneisen Constant in Liquid<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi mathvariant="normal">He</mml:mi></mml:mrow><mml:mprescripts/><mml:mrow/><mml:mrow><mml:mn>4</mml:mn></mml:mrow><mml:mrow/><mml:mrow/></mml:mmultiscripts></mml:mrow></mml:math>

By measuring the pressure dependence of the velocity of sound, we have determined both the pressure dependence of the density and the Gr\"uneisen constant $u$ of liquid $^{4}\mathrm{He}$. Measurements were made below 0.1 K and in the vicinity of 0.5 K. Our determinations of the pressure dependence of the density agree quite well with that determined by Boghosian and Meyer, who used a capacitance bridge. Since the latter results rely on the validity of the Clausius-Mossotti relation and a pressure-independent electric polarizability, the present work can be interpreted as supporting both of these assumptions. We found that $u({\ensuremath{\rho}}_{0})\ensuremath{\equiv}(\frac{\ensuremath{\rho}}{c})\frac{\mathrm{dc}}{d\ensuremath{\rho}}=2.84$ under the vapor pressure at 0.1 K. Using this value of $u$ to calculate the attenuation of sound according to a three-phonon mechanism, we obtain an attenuation of less than half the measured value. Thus, the present theory of sound attenuation must be incomplete.

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