Thesis (M.S. in Physical Oceanography) Naval Postgraduate School, September 1999. / "September 1999". Thesis advisor(s): Robert H. Bourke, James H. Wilson. Includes bibliographical references (p. 107-110). Also Available online.

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In the open ocean acoustic scattering by SONAR sources can be dominated by large fish schools. Multiple scattering effects are strong and the individual fish air-filled swimbladders scatter in the 1-10 kHz frequency range for most fish sizes. Furthermore, these schools are typically large in comparison to the acoustic wavelength and the individual fish typically swim in nearly periodic arrangements with a separation distance of approximately one body length. Hence, this work takes the perspective that fish schools can be studied simply and effectively by invoking the formalism of Bloch waves in periodic media. Analysis of the periodic school is aided through the Bloch theorem which reduces the study of the entire school to the study of a unit cell containing a single fish swimbladder. Application of the Bloch formalism to the school requires study of acoustic reflection from a semi-infinite half-space composed of an infinite tessellation of air-filled swimbladders in water. This media is denoted a fluid phononic crystal (PC). The reflection is considered, using a finite element discretization of the unit cell and an expansion of Bloch waves for the transmitted wave field. Next, scattering from a large finite school is studied through the context of the Helmholtz-Kirchhoff integral theorem where the semi-infinite PC pressure, determined by the Bloch wave expansion, is used as the surface pressure. Validation of results is accomplished via comparison with a finite element model (two dimensions) and a low frequency analytical multiple scattering model (three dimensions). Analysis of the dispersion relationship of the infinite PC yields useful information for a large school, namely, the frequency corresponding to target strength peaks, even as wave incidence angles and internal fish spacing are varied. The scattering effects attributed to the shape and weak internal disorder of the finite school were investigated with the surface integral method and a perturbation scheme. A general model using Bloch formalism, that encompasses the internal fish structure, fish biologic properties, and realistic school effects such as varying school geometry and disorder, was formulated. Transient analysis of the frequency dependent scattering, using the proposed approach developed in this thesis, may assist SONAR operators better classify large fish schools based on the observed characteristics of the scattered field.

The problem of two-dimensional acoustic scattering of an incident plane wave by a semi-infinite lattice is solved. The problem is first considered for sound-soft cylinders whose size is small compared to the wavelength of the incident field. In this case the formulation leads to a scalar Wiener--Hopf equation, and this in turn is solved via the discrete Wiener--Hopf technique. We then deal with a more complex case which arises either by imposing Neumann boundary condition on the cylinders' surface or by increasing their radii. This gives rise to a matrix Wiener--Hopf equation, and we present a method of solution that does not require the explicit factorisation of the kernel. In both situations, a complete description of the far field is given and a conservation of energy condition is obtained. For certain sets of parameters (`pass bands'), a portion of the incident energy propagates through the lattice in the form of a Bloch wave. For other parameters (`stop bands' or `band gaps'), no such transmission is possible, and all of the incident field energy is reflected away from the lattice.

Abstract The dissertation considers an inverse acoustic obstacle scattering problem in which the incident field is generated by a point source and the measurements are made in the near field region. Three methods to solve the problem of reconstructing the support of an unknown sound-soft or sound-hard scatterer from the near field measurements are presented. Methods are modifications of Kirsch factorization and modified Kirsch factorization methods. Numerical examples are given to show the practicality of one of the methods.

A poroelastic finite element formulation originally derived for modeling porous absorbing material in air is adapted to the problem of acoustic scattering from a poroelastic seafloor with a one-dimensional randomly rough interface. The developed formulation is verified through calculation of the plane wave reflection coefficient for the case of a flat surface and comparison with the well known analytical solution. The scattering strengths are then obtained for two different sets of material properties and roughness parameters using a Monte Carlo approach. These numerical results are compared with those given by three analytic scattering models---perturbation theory, the Kirchhoff approximation, and the small-slope approximation---and from those calculated using two finite element formulations where the sediment is modeled as an acoustic fluid. / text

In this thesis, we use asymptotic methods to solve problems of wave propagation through infinite and finite (only consider those that are finite in one direction) arrays of scatterers. Both two- and three-dimensional arrays are considered. We always assume the scatterer size is much smaller than both the wavelength and array periodicity. Therefore a small parameter is involved and then the method of matched asymptotic expansions is applicable. When the array is infinite, the elastic wave scattering in doubly-periodic arrays of cavity cylinders and acoustic wave scattering in triply-periodic arrays of arbitrary shape rigid scatterers are considered. In both cases, eigenvalue problems are obtained to give perturbed dispersion approximations explicitly. With the help of the computer-algebra package Mathematica, examples of explicit approximations to the dispersion relation for perturbed waves are given. In the case of finite arrays, we consider the multiple resonant wave scattering problems for both acoustic and elastic waves. We use the methods of multiple scales and matched asymptotic expansions to obtain envelope equations for infinite arrays and then apply them to a strip of doubly or triply periodic arrays of scatterers. Numerical results are given to compare the transmission wave intensity for different shape scatterers for acoustic case. For elastic case, where the strip is an elastic medium with arrays of cavity cylinders bounded by acoustic media on both sides, we first give numerical results when there is one dilatational and one shear wave in the array and then compare the transmission coefficients when one dilatational wave is resonated in the array for normal incidence. Key words: matched asymptotic expansions, multiple scales, acoustic scattering, elastic scattering, periodic structures, dispersion relation.

This thesis is concerned with the study of transient scattering of acoustic waves by an obstacle in an infinite domain, where the scattered wave is represented in terms of time domain boundary layer potentials. The problem of finding the unknown solution of the scattering problem is thus reduced to the problem of finding the unknown density of the time domain boundary layer operators on the obstacle’s boundary, subject to the boundary data of the known incident wave. Using a Galerkin approach, the unknown density is replaced by a piecewise polynomial approximation, the coefficients of which can be found by solving a linear system. The entries of the system matrix of this linear system involve, for the case of a two dimensional scattering problem, integrals over four dimensional space-time manifolds. An accurate computation of these integrals is crucial for the stability of this method. Using piecewise polynomials of low order, the two temporal integrals can be evaluated analytically, leading to kernel functions for the spatial integrals with complicated domains of piecewise support. These spatial kernel functions are generalised into a class of admissible kernel functions. A quadrature scheme for the approximation of the two dimensional spatial integrals with admissible kernel functions is presented and proven to converge exponentially by using the theory of countably normed spaces. A priori error estimates for the Galerkin approximation scheme are recalled, enhanced and discussed. In particular, the scattered wave’s energy is studied as an alternative error measure. The numerical schemes are presented in such a way that allows the use of non-uniform meshes in space and time, in order to be used with adaptive methods that are based on a posteriori error indicators and which modify the computational domain according to the values of these error indicators. The theoretical analysis of these schemes demands the study of generalised mapping properties of time domain boundary layer potentials and integral operators, analogously to the well known results for elliptic problems. These mapping properties are shown for both two and three space dimensions. Using the generalised mapping properties, three types of a posteriori error estimators are adopted from the literature on elliptic problems and studied within the context of the two dimensional transient problem. Some comments on the three dimensional case are also given. Advantages and disadvantages of each of these a posteriori error estimates are discussed and compared to the a priori error estimates. The thesis concludes with the presentation of two adaptive schemes for the two dimensional scattering problem and some corresponding numerical experiments.

Doctor of Philosophy / Department of Mechanical and Nuclear Engineering / Liang-Wu Cai / Acoustic scattering by a cylindrical scatterer comprising isotropic acoustic and orthotropic elastic layers is theoretically solved. The orthotropic material is used for the scattering problem because the sound speeds along radial and tangential axes can be different; which is an important property for acoustic cloaking design. A computational system is built for verifying the solutions and conducting simulations. Scattering solutions are obtained based on two theoretical developments. The first one is exact solutions for elastic waves in cylindrically orthotropic elastic media, which are solved using Frobenius method. The second theoretical development is a set of two canonical problems for acoustic-orthotropic-acoustic media. Based on the two theoretical developments, scattering by three specially selected simple multilayer scatterers are analyzed via multiple-scattering approach. Solutions for the three scatterers are then used for solving a “general” multilayer scatterer through a recursive solution procedure. The word “general” means the scatterer can have an arbitrary number of layers and each layer can be either isotropic acoustic or orthotropic elastic. No approximations have been used in the process. The resulting analytically-exact solutions are implemented and verified. As an application example, acoustic scattering by a scatterer with a single orthotropic layer is presented. The effects on the scattering due to changing parameters of the orthotropic layer are studied. Acoustic scattering by a specially designed multilayer scatterer is also numerically simulated. Ratios of the sound speeds of the orthotropic layers along r and θ directions are defined to satisfy the requirement of the Cummer-Schurig cloaking design. The simulations demonstrate that both the formalism and the computational implementation of the scattering solutions are correct.