Effects of bubbly layers on wave propagation

The transmission and reflection of an acoustic wave through a bubbly layer are investigated. Nonlinear model equations for the bubbly liquid are used. These equations first linearized and solved exactly for a time-harmonic incident wave. Then, numerical solutions of the nonlinear system are found. It is found that even for a small amplitude incident pressure wave, it is possible to have nonlinear transmitted and reflected waves. The limit where the bubbly layer is thin relative to the incident wavelength is also considered. By using the method of matched asymptotic expansions, it is found that the bubbly layer can be replaced by an interface subject to the continuity of pressure and an effective nonlinear jump condition. The latter involves the internal effects of the layer. Solutions of this limiting case are compared with the numerical results and good agreement is found even when the ratio of the bubbly layer thickness to the incident wavelength is of order 1.