Acoustic scattering by a pair of spheres

Abstract

If acoustic scattering by a single sphere is the most basic problem of scalar scattering, then sound scattering by a pair of spheres is next in the hierarchy of complexity. The problem has been formulated by several approaches in the past, but no actual detailed studies have been openly published so far. Two spheres insonified by plane waves at arbitrary angles of incidence are considered. The solution of this simplest of multiple-scattering problems is generated by exactly accounting for the interaction between the two spheres, which can be strong or weak depending on their separation, compositions, frequency, and directions of observation. The tools to attack this type of problem are the (forward/backward) addition theorems for the spherical wave functions, which permit the field expansions—all referred to the center of one of the spheres—by means of Wigner (3-j) symbols. The fields scattered by each sphere are obtained as pairs of (double) sums in the spherical wave functions, with coefficients that are coupled through an infinite set of two linear, complex, algebraic equations. These are then solved (by truncation) and used to obtain (i) the scattered fields and (ii) the scattering cross section of the pair of spheres. These exact results are illustrated with many plots of the form functions at various relevant incidence angles, separations, frequencies, etc. Finally, some asymptotic approximations for this problem that are analytically simple are obtained. They are displayed and compared to the exact solutions found above, with quite satisfactory results, even for the simple approximations used here. Thus the phenomenon is described, explained, graphically displayed, physically interpreted, and reduced to a simple accurate approximation in some important cases.

Scattering of Plane Waves by a Rigid Sphere in an Acoustic Quarterspace

“…In case of the sphere in the acoustic halfspace, there are strong oscillations in the form-function plot (triangular markers) that seem to be centered at a mean value of two with an oscillating amplitude of value near two (2) . This, as one would expect, is twice the value for a single sphere (circular markers) which would be the prediction offered by Born approximation in this case (12) . In particular, the amplitude maxima of the form functions Fig.…”

“…As the separation grows, the form function peaks become sharper and more densely packed. This behaviour may be clarified as follows (12) . The successive interactions among the various reflected and creeping waves in the acoustic quar- Fig.…”

Acoustic Radiation From a Pulsating Spherical Cap Set on a Spherical Baffle Near a Hard/Soft Flat Surface

“…The fluid is assumed to be inviscid and ideal compressible that cannot support shear stresses making the state of stress in the fluid purely hydrostatic. In view of the fact that the spherical cap is supposed to undergo time-harmonic surface pulsations, with frequency , the field equations may conveniently be expressed in terms of a scalar velocity potential as [21]: (1) where is the ambient fluid density, is the fluid particle velocity vector, is the acoustic pressure in the inviscid fluid, is the acoustic wave number, is the ideal speed of sound, and we have assumed harmonic time variations throughout with dependence suppressed for simplicity. Undoubtedly, the sound field radiated by a source may often be appreciably affected by a neighboring surface.…”

“…References [1] and [2] have each employed distinct analytical methods to examine acoustic scattering of plane compressional waves by two identical rigid and elastic spheres, respectively. The method of images in combination with the translational addition theorems for the spherical wave functions are extensively employed to study acoustic scattering by a hard spherical body near a hard flat boundary [3], by a thin spherical shell near a free (pressure release) surface [4], and by an ideal air-bubble near the sea surface [5].…”

Numerical modelling of multiple scattering between two elastical particles

“…When the multiple scattering term can be neglected (which is the case for d > Sa, approximately, as indicated by our experiments), the interaction between the two spheres can be calculated using the phase delay approach in agreement with equations (3)(4). In practice it means that a model consisting of many particles of equal size with an average separation distance of d = 4a represents a suspension of sediments with a concentration of 7% by volume, while the same model, but with a particle separation of d = 10a is equivalent to a suspension with 0.1% concentration by volume, or 3 -4 gll by weight.…”