Waves in Bubbly Liquids

Consider a gas-bubble liquid mixture with β the gas volume fraction p the pressure and ρ the density of the mixture. Let c_eff be the effective sound speed of the mixture and τ = ρ−1 the specific volume. We have that (1.1)$$ {\text{c}}_{{{\text{eff}}}}^{2} = \frac{{{\text{dp}}}}{{{\text{d}}\rho }} = \frac{1}{{{\text{d}}\rho {\text{/dp}}}} = \frac{1}{{{\text{d}}{{\tau }^{{ - 1}}}{\text{/dp}}}} = \frac{1}{{\kappa \rho }} $$ where the compressibility κ is defined by (1.2)$$ \kappa {\text{ = }}\frac{{ - 1}}{\tau }{\mkern 1mu} \frac{{{\text{d}}\tau }}{{{\text{dp}}}} $$ i.e. the change of volume with respect to pressure. Now let us assume that density and compressibility of the mixture are simply the averages over the two component values (1.3)$$ \rho {\text{ = }}{{\rho }_{{\text{g}}}}\beta {\text{ + }}{{\rho }_{\ell }}({\text{1 - }}\beta ),{\mkern 1mu} \kappa {\text{ = }}{{\kappa }_{{\text{g}}}}\beta {\text{ + }}({\text{1 - }}\beta ){{\kappa }_{\ell }} $$ where subscripts denote liquid or gas. The density of the gas is typically 1000 times smaller than that of the liquid while the compressibility of the liquid is negligible. Combining (1.1) and (1.3) with this simplification gives the formula (1.4)$$ {\text{c}}_{{{\text{eff}}}}^{2} = \frac{{\kappa _{{\text{g}}}^{{ - 1}}}}{{{{\rho }_{\ell }}\beta ({\text{1 - }}\beta )}} $$ If now p = const. ργ for the gas with γ the ratio of specific heats, we have κ_g−1=γp and hence (1.5)$$ {\text{c}}_{{{\text{eff}}}}^{2} = \frac{{\gamma \rho }}{{{{\rho }_{\ell }}\beta (1 - \beta )}} $$