Propagation and modulation of nonlinear spherical waves emitted by oscillating bubbles

In the numerical simulation of acoustic waves propagating in a liquid as described by the nonlinear Westervelt equation, it is typically assumed that both the wave emitting source and the background medium are at rest. In the present work, the nonlinear acoustic waves emitted at the moving bubble wall of an oscillating bubble are simulated by the means of suitable coordinate transformations of the Westervelt equation under the assumption of spherical symmetry and including the motion of the background medium as a result of the bubble oscillations. It is demonstrated that the motion of the wave emitting bubble wall and the background medium can cause nonlinear modulations of the wave amplitude and the rate of wave steepening. To this end, the finite difference wave solver is coupled to a Rayleigh-Plesset type model to predict the liquid pressure at the wall of a periodically excited gas bubble. Detailed investigations of the liquid pressure spectra at the bubble wall, predicted by the Rayleigh-Plesset type model, and at some distances from the bubble wall, predicted by the wave solver, are presented.