Acoustic Interactions with Submerged Elastic Structures

The following sections are included:

• Foreword

• Preface

• Contributors

• Contents

The extended boundary condition approach to acoustic and elastic wave scattering is formulated for inclusions obeying thin shell theory. Results are presented for the scattering of plane waves from infinite cylinders of elliptical cross section for ratios of major to minor axis varying from 1.0 (circular) to 2.0. Calculations were made for cylinders of plastic and steel and for elastomers having three different shear moduli. At low frequencies, both acoustic and elastic scattering by an elliptical shell are dominated by bending resonances. At higher frequencies, the scattering is strongly dependent on the angle of incidence, material characteristics, and the predominately extensional structural resonances. In an elastic material, resonant modal responses of the shell were found to increase mode conversion, of dilatational waves into shear waves. By comparison with elasticity theory for shells and by extrapolations from thin and thick plate theories, the limitations of the thin shell theory are shown to be material dependent.

The spherical, cylindrical and plane vector wave functions in elastodynamics of homogeneous and isotropic media are reviewed. This includes their trans-lational and rotational behaviour, the Green tensor expansions and the Betti identities (a type of orthogonality relations), as well as the transformations between the wave function systems. Three examples illustrate the usefulness of the given formulas. The first two are concerned with ultrasonic nondestructive testing and the formulas are used to transform the field from the ultrasonic probe to a form that is convenient for the scattering by a bounded defect, e.g. a crack, and a side-drilled hole, respectively. The last example is a multiple scattering problem with a sphere, a cylinder and a plane.

The reciprocity theorem for transient acoustic wavefields is taken as the point of departure for developing computational methods to model such wavefields. Mathematically, the theorem is representative of any ‘weak’ formulation of the wavefield problem. Physically, the theorem describes the ‘interaction’ between (a discretized version of) the actual wavefield and a suitably chosen ‘computational state’. The choice of the computational state determines which type of computational method results from the analysis. It is shown that the finite-element method, the integral-equation method and the domain integration method can be viewed upon as particular cases of discretizing the reciprocity relation. The local field representations of the acoustic pressure and the particle velocity in terms of nodal- and face-element expansion functions, respectively, are worked out in some detail. The emphasis is on time-domain methods. The relationship with complex frequency-domain methods is indicated.

We present the fundamentals of the linear theory of viscoelasticity. We review the classical approaches based on integral and differential constitutive equations, which lead to continuous and discrete time spectra, respectively. Particular attention is paid to the so-called fractional viscoelastic models based on differential equations of fractional order. In this respect we find it convenient to give an introduction to the fractional calculus.

In this chapter we review the main aspects of linear wave propagation in homogeneous, semi-infinite, viscoelastic media. In particular, we consider the so-called transient waves, thus named since they are generated by impact on an initially quiescent medium. The utility of the Laplace transform technique is pointed out in dealing with the evolution of these waves. We also discuss the concept of signal velocity and we provide wave-front expansions suitable for numerical computation.

The finite-element technique has been used to give very accurate numerical solutions to problems involving acoustic wave propagation in inhomogeneous fluid and anelastic media. These finite-element computer models can handle range and depth dependency in both density and sound and shear wave speeds, as well as rapid variations in bottom topography. This chapter describes the development and validation of these models, and their application to acoustic and elastic wave propagation in complex ocean environments.

Several investigators have demonstrated that measured mode nonorthogonality, among other factors, can make processing of shallow water vertical array data by use of modal compositions an improvement over processing in the space of the original pressure measurements. The potential for improvement using this technique is greatest when the Donorthogonality of the measured modes is largest, although the success of the transformation to modal amplitude space is governed by the condition of the transformation matrix. We have cast the Helmboltz equation for the Pekeris model in dimensionless form and studied the amount by which the measured normal modes fail to be orthogonal for a variety of ocean bottoms, array lengths and discretizations, and array positions. Environments are selected to reflect sediment types characteristic of the continental shelf. The nonorthogonality is found to be substantial and to exceed previous estimates for some cases even when the entire water column is spanned by a dense array. We have found that matched field processing using modal composition is likely to offer the most improvement over conventional matched field processing for water depths and frequencies at and just above modal onsets, for sound speed ratios close to one, and for arrays which span only a fraction of the water column. For full water column span discrete arrays with one more phone than mode, there is a zone of orthogonality covering one third to one half the possible depth and frequency combinations for all cases studied.

Nearfield acoustical holography is a procedure for a high resolution reconstruction and display of the acoustic field. It consists of three step procedure: the measurement of the acoustic field over a hologram surface, the numerical reconstruction of the field over the image region and the computer dispay of the resulting reconstruction. The numerical reconstruction of the acoustic field is discussed in this work. This procedure requires the solution of an ill posed inverse problem requiring the inversion of an ill-conditioned operator. For separable geometries this is accomplished by decomposing the acoustic field into an orthogonal set of functions obtained by the use of separation of variables and truncating the series. Nonseparable geometries are treated numerically using the boundary element formulation and either singular value decomposition or eigenfunction expansion techniques to invert the operator.

The method of moments technique is applied here to the problem of acoustic scattering from cylinders of arbitrary cross section embedded in a fluid. It is used to eliminate internal resonance effects in the scattering problem.

Problems for transient sources in a multilayered elastic medium may be treated by the theory of generalized ray. In this theory the Laplace-transformed multiply reflected and/or transmitted cylindrical/spherical wave, known as a ray integral, is constructed by linear superposition of the Laplace-transformed plane waves. The potential reflection and transmission coefficients for plane waves are basic to such a superposition. The well-known potential reflection and transmission coefficients are applicable only to the formulations of the theory of generalized ray for which the total reflected/transmitted wave motion is constructed by linear superposition of the two-dimensional plane waves, the coupled P and SV waves, or the SH waves. These formulations were developed for the following problems: (1) plane strain (two-dimensional) problems in the Cartesian coordinates (e.g., a line of vertical forces emitting the cylindrical P and SV waves), (2) antiplane strain (two-dimensional) problems in the Cartesian coordinates (e.g., a line of horizontal forces emitting the cylindrical SH wave), (3) axisymmetric (three-dimensional) problems in the cylindrical coordinates (e.g., a vertical force emitting the spherical P and SV waves, or a torque with vertical axis emitting the spherical SH wave), and (4) asymmetric (three-dimensional) problems in the cylindrical coordinates (e.g., an oblique force emitting the spherical P, SV, and SH waves). In the present paper we develop the theory of generalized ray for two three-dimensional problems in the Cartesian coordinates, the asymmetric point load source in a horizontally layered elastic medium, and the same source in a dipping elastic layer overlying an elastic half-space. For the two problems, the total wave motion at an interface is composed of the coupled spherical P and S waves, and the multiply reflected and/or transmitted ray integrals are therefore constructed by linear superposition of the Laplace-transformed three-dimensional coupled plane P and S waves. The potential reflection and transmission coefficients for the three-dimensional coupled plane P and S waves are basic to such a superposition. The inversion of the ray integrals is treated by the modified Cagniard-de Hoop method. A discussion of the numerical integration of the ray integrals together with numerical examples is presented in the paper. The paper also discusses how a transient response of a viscoelastic half-space perturbed by the buried asymmetric point load source can be obtained from the solution of the elastic problem by the use of the numerical version of the elastic-viscoelastic correspondence principle.

This article is concerned with the nonlinear analysis of the stability of thick elastic bodies subjected to finite elastic deformations. The analysis is based on the theory of small elastic deformations superimposed on a large elastic deformation. Attention is drawn to methods developed in the stability analysis of fluids and of thin shells and plates which are readily applicable to the present circumstances. The state of development of the nonlinear stability analysis of thick elastic bodies is summarized in order to provide a basis for subsequent studies, and some new results relating to the stability of an elastic plate subjected to a pre-stress associated either with uniaxial thrust or with simple shear in the presence of all-round pressure are discussed.

Near-neutral modes in the neighbourhood of so-called neutral configurations are considered to depend on, for example, a slow time variable, and nonlinear evolution equations for the mode amplitudes are derived both in the case of a monochromatic mode and for a resonant triad of modes. The crucial role of the ‘nonlinear coefficient’ in such an equation in the analysis of stability, imperfection sensitivity and localization is highlighted. An efficient (virtual work) method for the determination of this coefficient is described together with an alternative method based on the calculation of the total energy of a monochromatic near-neutral mode. The influence of the boundary conditions and of the form of the pre-stress is examined and explicit calculation of the nonlinear coefficient is provided for the two representative pre-stress conditions mentioned in the above paragraph. It is shown, in particular, that a resonant triad of modes has an effect similar to that generated by the presence of a geometrical imperfection.

The Appendices gather together for reference certain expressions which are used in the body of the article. These include expressions, not given previously in the literature, for the components of the tensor of third-order instantaneous elastic moduli in terms of the principal stretches of the deformation in respect of a general form of incompressible isotropic elastic strain-energy function.

The following sections are included:

• PROFESSOR HERBERT ÜBERALL'S PUBLICATIONS IN ACOUSTICS, ELASTODYNAMICS AND ELECTROMAGNETICS

• LIST OF BOOKS WHICH PROFESSOR H. ÜBERALL AUTHORED, CO-AUTHORED, EDITED OR CO-EDITED

• STUDENTS WHO GRADUATED UNDER PROFESSOR HERBERT ÜBERALL FROM THE CATHOLIC UNIVERISTY OF AMERICA

• Author Index

• Subject Index

• About the Editors