Sound scattering by bubble clouds near the sea surface

The classical exact formulation required to evaluate the form function (or the scattering cross-section, SCS) of a single, ideal, air bubble in a boundless liquid is briefly recalled. It is then immediately generalized to the case of a round cloud of many possibly interacting such bubbles of known volume concentration, contained within the same boundless medium. This is further generalized to the case when the bubble cloud is near a free surface. The presence of the nearby pressure release surface, assumed flat, substantially alters the cloud's scattering cross-section relative to its value in the absence of boundaries. We then use an earlier technique of ours [i.e., see I.E.E.E. J. Ocean. Eng. 20, 285-293 (1995)] based on the method of images that uses the addition theorem for the spherical wave functions, to relate all the scattered sound fields to a common origin and thus obtain the (modified) SCS of the cloud now near the boundary. This formulation accounts for all orders of multiple scattering and yields an infinite set of coupled algebraic equations for the coupling coefficients. This set is then solved for the coupling coefficients in terms of infinite sums of products of pairs of Wigner 3-j symbols, which are then used to construct and evaluate the form function. We display numerical results in four cases that correspond to geographical sites in which the bubble concentrations within the cloud have been measured along a couple of oblique upward directions, or have been assumed to have increasing (and in a few instances, purposely unrealistically high) values. In all cases considered here the bubble clouds are only a few meters beneath the sea surface and consist of ideal bubbles. The results are also compared to those found in the absence of a boundary in all the cases considered.