Sound scattering by a spherical object near a hard flat bottom

Abstract

Ahstract-We consider the scattering of plane acoustic waves by spherical objects near a plane hard surface. The angl£'$ of incidence are arbitrary and so are the distances of the objects from the hard boundary. We use the method of images. The final result for the sound field [ef. (21)] consists of four parts: the incident field and its refiection from the boundary, which are shown combined; the scattered field from the sphere, and that scattered by its image. These last two appear coupled since both sphere and image are repeatedly interacting with each other.The entire solution is referred to the center of the real sphere. This can be accomplished in an exact fashion by means of the addition theorems for spherical wave-functions. These theorems are taken from the atomic physics literature, where they are more frequently used. The required coupling coefficients, bmn, are obtained from the solution of an infinite linear complex system of transcendental equations with coefficients given by series. The system is suitably truncated to obtain numerical predictions for the form-functions by means of the Gauss-Seidel iteration method. Many calculations are displayed exhibiting the distortion that the proximity of the hard boundary causes on the free-space solution. The form-functions are graphed versus ka, for various values of the normalized separation D == d I a of the sphere from its image. They are also plotted versus the angle of observation, for fixed values of S! = ka and D. These plots are the exact benchmark curves against which the accuracy of approximate solutions, found by other methods, could be assessed. They could also serve to determine the distances above the bottom, beyond which the bottom effeet could be neglected. This is an idealized model to predict the distorted sonar cross section of a hard spherical object near a hard fiat bottom. L I NTRODUCTION To STUDY acoustic scattering bv an object near an interface or by many objects placed in various spatial configurations, the starting point is the addition theorems. Such theorems allow us to considcr thc scattered fields produced by the various bodics, or by the body and its image (or images), all referred to a common origin. A number of works lIJ-[7] have dealt with this subject. The use of these theorems in physical cases leads to Wigner (3 -j) symbols. In their quantum mechanical or atorn.ic physics context, these computational tools are discussed in many works [8]- [14]. Their use in classical physics is straightforward. Scattcring by two or more hodies which are relatively close to each other, and thcir interaction is significant, is essentially a multiple scattering problem. The literat ure on this related topic is also nontrivial Manuscript

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

“…Examination of these plots leads to the following important observations. 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) .…”

“…Seybert and Soenarko (1) applied the boundary integral equation (BIE) method to study general acoustic radiation and scattering in a three-dimensional halfspace. Gaunaurd and Huang (2) employed the translational addition theorems for the spherical wave functions to study acoustic scattering by a hard spherical body near a hard flat boundary. They also considered acoustic scattering by a thin spherical elastic shell near a free surface (3) , and by an ideal air-bubble near the sea surface (4) .…”

Sound Radiation of a Pulsating Sphere in the Outlet of a Hard/Soft Semi-Spherical Cavity in a Flat Screen

“…Thompson Jr. (1973;1976) considered an interesting problem of sound radiation by a sphere eccentrically located within a fluid sphere using the Clebsch-Gordan coefficients, which enables expressing the acoustic field of a spherical source in some eccentric spherical coordinates, and also solved the problem of sound radiation by a spherical source located near a fluid sphere. The method was further developed by Gaunaurd and Huang (1996), who used it for solving the problem of sound scattering by a spherical object near a hard flat bottom. Hasheminejad (2003) analyzed rigorously the modal radiation load on a sphere immersed in an acoustic halfspace bounded by a rigid plane using the same method.…”

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

“…It is clear that the proximity of the wall makes the problem more difficult to solve. If the wall is initially idealized as rigid, planar, and of infinite extent, a very simple theoretical device known as the method of images can smoothly take its presence into account [3]- [5], [21]. This method substitutes the original boundary value problem by one with two sources in an unbounded medium (i.e., the original source and the mirror image source).…”

“…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]. Axisymmetric acoustic radiation from a spherical source vibrating with an arbitrary, time-harmonic velocity distribution while positioned wholly outside a fluid sphere is examined in [6].…”