Numerical analysis of multiple scattering of an acoustic wave on a set of sound-permeable spheres in 3D space

In the study of the scattering of an acoustic wave by a set of spherical obstacles of small sizes, one of the most important problems is to determine the main characteristics of this phenomenon, including the total scattering cross-section. Knowledge of the characteristics allows the most complete interpretation of the numerical results obtained in the study of multiple scattering effects on small obstacles. A review of the scientific literature showed that today all theoretical and numerical studies are devoted to the systems consisting of one or two scatterers, or are limited by the cases that reduce the scattering problem on a set of spheres to scattering on a single two-phase region or ignore the backscattering between neighboring scatterers; the interaction between spherical obstacles cannot be completely taken into account in these limiting cases. The main purposes of this work are to derive an explicit formula for the total scattering cross section on a set of interacting sound-permeable spheres and to conduct on this basis a numerical analysis of multiple scattering on systems of spheres located in so-called basic configurations. Such a formula was obtained using the summation theorems for spherical wave functions. It is applicable to any number of spheres of different radii freely located in 3D space in the presence of an arbitrary external sound field. Computational experiments were performed for the systems consisting of: two spheres located at the same distance from a monopole radiation source, three spheres located in four basic configurations, and identical spheres of a 11×11 planar uniform configuration. It was assumed that all these systems are subjected to a spherical wave from a monopole radiation source. As a result of studying the total scattering cross-section, with and without taking into account the interaction between the spheres, which was performed at varying system parameters (the density and speed of sound around and inside the spheres, the external field frequency, the distance between the sphere centers, and the location of the spheres relative to each other), it was possible to identify a parameter space where the effects of multiple scattering cannot be neglected.