Giant monopole resonances in the scattering of waves from gas-filled spherical cavities and bubbles

Abstract

Continuing our previous work on viscoelastic scattering of p waves from fluid-filled spherical cavities in sound-absorbing materials, we now perform the theoretical analysis needed for the complete study of the monopole mode of vibration. Our approach is based on the Breit–Wigner formulation of nuclear scattering theory, and from it we show how the scattering amplitudes consist of two parts, the smooth background of the evacuated cavity and the resonance contribution of the filler fluid. We present a complete analytical study of the giant fundamental resonance and its overtones in the monopole case. The results are then particularized to the Rayleigh region and also to the case of evacuated cavities in lossless rubber and to gas bubbles in water, recovering and extending previous results available in the literature. Studying the sound pressure level at the center of air-filled cavities in lossy materials, we quantitatively determine how absorption shifts the resonance peaks and broadens their width. Using our theory we find that the width of the resonance peaks depends linearly on the resonance frequencies, and we show how this important result can be used to determine the shear absorption parameter of viscoelastic materials. We also find that the giant monopole resonance for air-filled cavities in rubber contains a contribution from the shear waves in the cavity wall and another from the resonance oscillations of the air-filling in a 15:1 ratio, for this particular material combination. We display numerous plots, some obtained from the analytic monopole formulas developed here, and some numerically obtained by computer for dipoles and quadrupoles. Finally, we show graphs of the summed scattering amplitudes fpp and fps accounting for the first twenty multipole contributions, and the dominance of the monopole term emerges again from this result. We have assumed ’’weak’’ absorption to simplify the analytical treatment, but any viscosity level can be handled by computer using this method.