Scattered Pressure Field Research Articles

Boundary‐conforming coordinates with grid line control for acoustic scattering from complexly shaped obstacles

AbstractThe current work sets forth a practical approach to numerically solve two‐dimensional direct acoustic scattering problems from complexly shaped scatterers with severe singularities, such as corners and cusps. First, boundary conforming coordinates are generated. This generation is performed through an elliptic grid generator algorithm, including control of the coordinate lines. The grid line control solely depends on the initial distribution of grid points. Following the grid generation process, the initial boundary value problem, modelling the scattering phenomenon, is formulated in terms of the new curvilinear coordinates, and a finite‐difference time domain method is implemented. The presence of the boundary singularities causes instability of the numerical method. However, by appropriately controlling the distance between grid lines in the vicinity of these singularities, stability and convergence are achieved. A semianalytical formula for the differential scattering cross‐section is obtained from the discrete Fourier transform of the computed scattered pressure field. The method is successfully applied to several interesting scatterers of various shapes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Acoustic scattering from two spheres, one with a small radius

The scattering of a plane acoustic wave from an acoustically penetrable or impenetrable (soft or hard) sphere separated at a distance from another sphere, also penetrable or impenetrable (soft or hard), of acoustically small radius, is examined. The penetrable spheres and the surrounding medium are fluids or fluidlike; i.e., they do not support shear waves. Separation of variables, in conjunction with translational addition theorems for spherical wave functions, is used. Analytical expressions are obtained for the scattered pressure field and the scattering cross sections. Numerical results are given for penetrable and impenetrable spheres, showing the influence of the small sphere on the scattering cross sections of the other sphere.

Scattering of an obliquely incident plane wave from an immersed transversely isotropic rod covered by an isotropic cladding

Valuable information is contained within the scattered pressure field from a submerged elastic cylinder. Many theoretical and experimental investigations on the scattering from elastic cylinders have been carried out during the past few years. This includes recent work of the authors on modeling of the scattered pressure field from immersed transversely isotropic cylinders and immersed isotropic clad rods. In the current paper, an analytical solution to the problem of acoustic wave scattering from an immersed transversely isotropic cylinder covered by an isotropic cladding is presented. The mathematical expressions are derived for the far-field backscattered amplitude spectrum, normal modes, frequency spectrum, and phase diagrams resulting from oblique insonification of an infinite, transversely isotropic cylinder covered by an isotropic cladding. The validity of the mathematical model is verified by comparing the results with data available in the literature for simpler cases, such as transversely isotropic rods and isotropic clad rods. The results are also consistent with elasticity theory.

Further comments on the paper by Zinoviev and Bies, “On acoustic radiation by a rigid object in a fluid flow”

In this note, in response to the previous note by Zinoviev and Bies [Author's Reply to: F. Farassat, Comments on the paper by Zinoviev and Bies “On acoustic radiation by a rigid object in a fluid flow”, Journal of Sound and Vibration 281 (2005) 1224–1237], the present authors briefly discuss the assumptions used in the acoustic analogy and its intended applications. It is pointed out that the scattering problems discussed by Zinoviev and Bies do not fall within the intended applications of the Ffowcs Williams–Hawkings (FW–H) equation. However, the FW–H equation can be used to derive an integral equation for finding scattered pressure fields. We derive this integral equation and show the validity of the equation for some of the examples of Zinoviev and Bies. We respond to the other issues brought up by these authors in their notes. We believe that Zinoviev and Bies have misinterpreted the acoustic analogy and have not applied the Curle formula and the FW–H equation correctly.

Author's reply

In this note, in response to the previous note by Zinoviev and Bies [Author's Reply to: F. Farassat, Comments on the paper by Zinoviev and Bies On acoustic radiation by a rigid object in a fluid flow, Journal of Sound and Vibration 281 (2005) 1224-1237], the present authors briefly discuss the assumptions used in the acoustic analogy and its intended applications. It is pointed out that the scattering problems discussed by Zinoviev and Bies do not fall within the intended applications of the Ffowcs Williams-Hawkings (FW-H) equation. However, the FW-H equation can be used to derive an integral equation for finding scattered pressure fields. We derive this integral equation and show the validity of the equation for some of the examples of Zinoviev and Bies. We respond to the other issues brought up by these authors in their notes. We believe that Zinoviev and Bies have misinterpreted the acoustic analogy and have not applied the Curle formula and the FW-H equation correctly.

Sound scattering by a hard axisymmetric object in a double layered ocean

An acoustically hard object is placed in a double layered ocean very far from a point source of acoustic waves. The object is either floating in the water layer or buried in the sediment layer. Both layers are homogenous. The size of the object is small when compared to the depth of the water channel. The free surface of the sea is assumed to be soft and the bottom is assumed to be hard. Between the two layers the classical diffraction boundary conditions are taken. The deep water approximation method is being extended to provide an approximating analytical expression of the scattered pressure field.

Acoustic scattering by a circular cylinder parallel with another of small radius.

The scattering of a plane acoustic wave by an infinite penetrable or impenetrable circular cylinder, parallel with another one, also penetrable or impenetrable, of acoustically small radius, is considered. The method of separation of variables, in conjunction with translational addition theorems for cylindrical wave functions, is used. Analytical expressions are obtained for the scattered pressure field and the various scattering cross sections, for normal incidence. Numerical results are given for penetrable and impenetrable cylinders.

A T-MATRIX PERTURBATION FORMALISM FOR SCATTERING FROM A ROUGH FLUID-ELASTIC INTERFACE

A perturbative representation of the null field T-matrix for scattering a pressure wave from a fluid elastic interface with surface roughness is developed. It is shown that thenth order T-matrix may be calculated recursively and expressed in terms of the elements of a zeroth order T-matrix. Further, it is shown that the expression for thenth order T-matrix is general in form and may be specialized to scattering from rigid and soft rough surfaces and from the rough surface of a fluid sediment. The T-matrix for thenth order spectral amplitude of the scattered pressure field in the fluid is calculated and its diagrammatic representation is constructed. For sinusoidal surface roughness, the perturbative representation of the T-matrix is used to calculate the spectral amplitudes of the Floquet modes of the pressure field scattered in the fluid. The results are compared with those obtained from a non-perturbative representation of the T-matrix and the accuracy and region of applicability of the formalism is determined. Then the relative scattering amplitudes of some of the individual scattering processes that occur in the diagrammatic representation of the T-matrix are calculated through second order. It is shown that diagrams that correspond to scalar processes in which surface pressure fields are excited and propagated tend to dominate diagrams that correspond to vector and tensor processes in which surface displacement fields are excited and propagated. A “scalar” approximation to the perturbation series is defined in which only scalar diagrams in the perturbation series are maintained. Numerical results for the perturbation series in the scalar approximation to fourth order are compared with exact results and it is shown that the scalar approximation provides reasonably accurate results for some applications.

The multiple scattering interactions of cobbles and pebbles lying on the ocean floor

A new model for predicting the sonar target strengths of ensembles of rocks lying on the ocean floor has recently been presented. The model represents rocks by either rigid movable spheres or elastic spheres of variable size, density, and numerical/geometrical configuration. When the rocks are densely packed, and therefore in close proximity to each other, it is important to consider the contribution that multiple scattering interactions make to the collective scattering response. To investigate this question, a T-supermatrix formalism, a self-consistent procedure which includes all orders of multiple scattering [see Lim and Hackman, J. Acoust. Soc. Am. 91, 613–638 (1992)], has been developed to describe the radiatively coupled ensemble. The method allows the coherently summed scattered pressure field in any direction to be determined, as well as the total power scattered by the system. Application to rocks of cobble and pebble size classes indicates that the effect of multiple interactions is dependent upon the acoustic frequency, the rock spacing relative to the radius, and the grazing angle. The results obtained provide a basis for estimating the relative importance of these interactions in determining target strength. [This work was supported by ONR/NRL.]

Scattering from an elastic shell and a rough fluid–elastic interface: Theory

A null-field T-matrix formalism is developed for scattering a pressure wave from a stationary elastic shell immersed in a homogeneous and isotropic fluid half-space and in proximity to a rough fluid–elastic interface. Helmholtz–Kirchhoff integral representations of the various scattered pressure and displacement fields are constructed. The surface fields are required to satisfy the elastic tensor boundary conditions and the scattered fields are required to satisfy the extended boundary condition. Spherical basis functions are used to construct a free-field T-matrix for the elastic shell and rectangular vector basis functions are used to construct a representation of the free-field T- matrix for the rough fluid–elastic interface. The free-field T-matrices are introduced into the Helmholtz–Kirchhoff and the null-field equations for the shell-interface system and a general system of equations for the spectral amplitudes of the various fields is obtained. The general system of equations is specialized to scattering from periodic surface roughness and an exact solution for the scattered pressure field in the fluid is obtained. Then the general system of equations is specialized to scattering from small-amplitude arbitrary roughness profiles and a perturbative solution is obtained. It is shown that the formalism contains multiple scattering effects on the rough surface and between the rough surface and the shell.

High-frequency scattering from a target in proximity to a randomly rough interface

A T-matrix formalism for free-field target scattering and a Helmholtz–Kirchhoff integral equation in the Kirchhoff approximation for rough interface scattering are used to calculate high-frequency plane-wave scattering from a stationary target in proximity to an interface with random surface roughness. Scattering from targets located above and below the interface are considered: When the target is located above the interface, it is assumed that the scattered pressure field is simply the superposition of the target and rough interface scattered fields. When the target is located below the interface, it is assumed that target-interface multiple scattering can be ignored and that the scattered pressure field above the interface is given by the superposition of the field scattered from the interface and the field scattered from the target and transmitted through the interface when the field incident on the target is the transmitted plane-wave field. Numerical calculations indicate some of the effects of surface roughness and target physical and geometric parameters on the ability to ‘‘see’’ the target.

Non-linear transient analysis of submerged circular plates subjected to underwater explosions

A new coupled finite element and boundary integral formulation is developed for non-linear transient analysis of submerged thin circular plates subjected to underwater explosions. In this new formulation, the governing equations of motion of the structure are completely decoupled from those of wave propagation by applying Kirchhoff, s integral equation on the wet surface of the structure, thereby eliminating the need for modelling the surrounding fluid. Using Kirchhoff thin plate theory, an axisymmetric ring plate element is formulated, which takes into account both geometric and material non-linearities as well as strain-rate effects. The scattered pressure field due to the fluid-structural interaction is calculated by solving the surface integral equation in the context of element discretization method. The effects of water cavitation on structural response are included by using an appropriate pressure criterion. The time-dependent solution of the coupled fluid-structure system is then solved by applying a staggered solution algorithm at each time step in a direct time-integration procedure. The proposed formulation has been tested for a number of applications and the results obtained are compared with experiments. It is observed that the new formulation can provide reasonable solutions to both near- and far-field interaction problems for underwater shock response analysis.

Scattering from a fluid loaded elastic spherical shell in proximity to a rough interface: Numerical results.

A null field T‐matrix formalism is developed and used to calculate plane‐wave scattering from a fluid loaded elastic spherical shell in proximity to sound hard, sound soft, fluid–fluid, and fluid–elastic interfaces with periodic surface roughness. For each type of interface, the Helmholtz–Kirchhoff integral representation of the various scattered pressure and displacement fields are constructed; the surface fields are required to satisfy the appropriate boundary conditions and the scattered fields are required to satisfy the extended boundary condition. Spherical basis functions are used to construct a free field T‐matrix for the elastic shell and rectangular vector basis functions are used to construct a representation of the free field T‐matrix for the rough interface. The free field T‐matrices are introduced into the Helmholtz–Kirchhoff equations for the scattered fields and the null field equations for the shell‐interface system and an ‘‘exact’’ analytical solution is obtained. Numerical results are obtained that demonstrate the effects of boundary type, elastic parameters, roughness geometry, amplitude, and slope on the scattered pressure field and on the ability to ‘‘see’’ the scatter from the elastic shell.

Scattering of an obliquely incident wave by a transversely isotropic cylinder.

While the scattering of a plane acoustic wave from a solid isotropic cylinder has been extensively studied for the past several decades, very little has been investigated regarding scattering from anisotropic cylinders. In this paper, the mathematical formulation for the scattering of a plane acoustic wave incident at an arbitrary angle α on an infinite transversely isotropic cylinder is developed. The normal mode solution is based on decoupling of the SH wave from the P and the SV waves. The resulting partial differential equations have closed‐form solutions and the scattered pressure field can be calculated at any point outside the cylinder. The solution degenerates to the well‐known simple model for isotropic cylinders in the case of very weak anisotropy. The validity of the mathematical model is verified first by applying it to the familiar case of an isotropic aluminum cylinder. The model is then applied to a transversely isotropic cylinder generated by perturbing various elastic constants of the iso...

The computation of the scattered pressure field from a cylinder embedded between two half-spaces with different densities

In this Letter a straightforward, efficient method for computing the acoustic field scattered by a cylinder embedded halfway between two half-spaces is presented. These half-spaces have the same velocity but a different density. This method should be useful for benchmarking more general codes.

Acoustic scattering in three‐dimensional fluid media

The scattering of acoustic waves from three‐dimensional compressible fluid scatterers is considered. Particular attention is paid to cases where the scatterers have moderate magnitude in compressibility contrast and nondimensional wave number. The perturbation method based on Pade approximants developed by Chandra and Thompson [J. Acoust. Soc. Am. 92, 1047–1055 (1992)] is extended to allow the solution of problems in three dimensions. It is shown that the functional form afforded by the Pade approximant model allows one to represent and evaluate the characteristic resonances and mode shapes of the scattered pressure field. These modes are a function of the compressibility contrast and the frequency of the incident pressure wave. Numerical results are shown to compare favorably with analytical solutions for scattering from a sphere. The Pade approximant method is shown to be feasible for calculating the pressure scattered from an inhomogeneous distribution of scatterers as well as for determining internal ...

Acoustic scattering from a sphere of small radius coated by a penetrable one

The scattering of a plane acoustic wave from an impenetrable or a penetrable sphere of acoustically small radius coated by another penetrable sphere, is considered in this work. The method of separation of variables is used, combined with translational addition theorems for spherical wave functions. Analytical expressions are obtained for the scattered pressure field and the various scattering cross sections. Numerical results are given for various values of the parameters and for both an impenetrable or a penetrable inner sphere.

Acoustic scattering from a hemicapped cylindrical shell with substructures

This paper extends the study on acoustic radiation of axisymmetric submerged shells of finite length to the scattering problem. A modal-based method is developed to determine the scattering from finite shells insounified by plane incident waves. A surface variational principle provides the acoustic impedance relationship between the surface pressure and displacement fields, which are expanded into a series of shape functions. Lagrange multipliers are introduced in a Lagrange formulation of the dynamic analysis to account for the interaction between the shell and internal substructures. Surface and far-field pressures scattered from the elastic shell are calculated. It is found that, at certain angles, the scattered pressure field has pronounced peaks, particularly as the incident frequency increases. A wave number domain analysis is performed on the surface displacement field. A dominant wave number is found which accounts for the high directivity pattern of the far field. The effect of substructures on the far field is also investigated.

The scattering of sound from a spatially distributed axisymmetric cylindrical source by a circular cylinder

General expressions for the scattering of sound from a spatially distributed cylindrical source by a circular cylinder are given. The incident field is determined using a Hankel transform. The scattered fields are determined by matching the pressure and radial velocity at the surface of the cylinder. Examples of total and scattered pressure fields are given for a cylindrical line source and a axisymmetric source with a Gaussian spatial distribution. An example is also given for a circular disk source.

The exact sonar cross section of an elastic spherical shell near the sea surface

The acoustic scattering by a submerged spherical elastic shell near a free surface and insonified by plane waves at arbitrary angles of incidence is analyzed in an exact fashion by the classical separation of variables method. To satisfy the boundary conditions at the free sea surface, as well as on the surfaces of the elastic shell, the mathematical problem is formulated using the method of images. The scattered wave fields are expanded in terms of spherical wave functions using the addition theorems. The final results come out in terms of Wigner 3‐j symbols or Clebsch–Gordan coefficients. Quite similar to the problem of scattering by multiple spheres, the computation of the scattered pressure field involves the solution of an ill‐conditioned complex matrix system, of size that depends on the number of terms in the modal series required for convergence. This, in turn, depends on the value of the frequency and on the proximity of the steel shell to the free surface. The matrix equation is solved by the Ga...