Low frequency sound scattering from spherical assemblages of bubbles using effective medium theory

The determination of the acoustic field scattered by an underwater assembly of gas bubbles or similar resonant monopole scatterers is of considerable theoretical and practical interest. This problem is addressed from a theoretical point of view within the framework of the effective medium theory for the case of spherically shaped assemblages. Although being valid more generally, the effective medium theory is an ideal instrument to study multiple scattering effects such as low frequency collective resonances, acoustically coupled breathing modes of the entire assembly. Explicit expressions for the scattering amplitude and cross sections are derived, as well as closed form expressions for the resonance frequency and spectral shape of the fundamental collective mode utilizing analytical S-matrix methods. This approach allows, in principle, a simultaneous inversion for the assembly radius and void fraction directly from the scattering cross sections. To demonstrate the validity of the approach, the theory is applied to the example of idealized, spherically shaped schools of swim bladder bearing fish. The analytic results of the theory are compared to numerical first-principle benchmark computations and excellent agreement is found, even for densely packed schools and frequencies across the bladder resonance.