High Frequency Nonlinear Waves in Bubbly Liquid

Abstract

Consider a monodisperse bubbly liquid, with volume concentration of bubbles ±₀, bubble resonance frequency ω ₀, and mean density ρ₀, and let c ₀, c _∞ stand for the sound speeds at zero and infinite frequency (equilibrium and frozen sound speeds), respectively. Also let Γ stand for an effective adiabatic exponent in a local approximation p p ₀ = (ρ/ρ₀)^Γ for the equation of state in pure water (Γ ≈ 7). Then (cf Crighton 1991 for a simple derivation) the equation for weakly nonlinear one-dimensional wave propagation through the mixture, at frequencies well above the resonance, is

$$\frac{1} {{c_\infty ^2 }}\frac{{\partial ^2 p}} {{\partial t^2 }} - \frac{{\partial ^2 p}} {{\partial x^2 }} + \frac{{\omega _0^2 }} {{c_0^2 }}p = \frac{{\left( {\Gamma + 1 - 2\alpha _0 } \right)}} {{2\left( {1 - \alpha _0 } \right)}}\frac{1} {{\rho _0 c_\infty ^4 }}\frac{{\partial ^2 p^2 }} {{\partial t^2 }},$$

(1.1)

where p is the pressure fluctuation. In terms of the velocity fluctuation u = p/ρ₀ c _∞, the reduction of this compound wave equation to an equation for a predominantly right-propagating wave is

$$\frac{\partial } {{\partial x}}\left( {\frac{{\partial u}} {{\partial t}} + c_\infty \frac{{\partial u}} {{\partial x}} + \frac{{\left( {\Gamma + 1 - 2\alpha _0 } \right)}} {{2\left( {1 - \alpha _0 } \right)}}u\frac{{\partial u}} {{\partial x}}} \right) - \frac{{\omega _0^2 c_\infty }} {{2c_0^2 }}u = 0.$$

(1.2)