Robert P. Gilbert

Homogenizing the acoustic properties of the seabed: part I

One of the pressing problems in underwater acoustics today is formulating and then solving a model for interaction of acoustic waves in a shallow ocean with the seabed. Shallow-water/seabed waveguide, direct and inverse wave propagation problems are ubiquitous in applied science and technology. One such application is for inverse imaging of objects submerged in the ocean or the seabed. As much of the acoustic energy passes into the seabed, this imagery is possible only if the sea environment (water, sediment, interfaces), in the absence of the object, is properly characterized beforehand. This means that a suitable model of the sediment and of propagation of sound therein must be developed and a method be proposed for solving the inverse problem of the identification of the mechanical parameters involved in this model. This model, as well as the sediment parameter and object identification scheme, must be able to take into account sound speed and density variations in the water as well as the behavior of sound in the seabed. In general, either an acoustic pulse, or a monochromatic signal with frequency o is used. Consequently, not only acoustic signals with acoustic frequencies spread about a central frequency, but time-harmonic solutions are of interest. There have been several acoustic models of the seabed [9,10,16]; however, the primary one in usage goes back

Application of the multiscale FEM to the modeling of cancellous bone

This paper considers the application of multiscale finite element method (FEM) to the modeling of cancellous bone as an alternative for Biot’s model, the main intention of which is to decrease the extent of the necessary laboratory tests. At the beginning, the paper gives a brief explanation of the multiscale concept and thereafter focuses on the modeling of the representative volume element and on the calculation of the effective material parameters, including an analysis of their change with respect to increasing porosity. The latter part of the paper concentrates on the macroscopic calculations, which is illustrated by the simulation of ultrasonic testing and a study of the attenuation dependency on material parameters and excitation frequency. The results endorse conclusions drawn from the experiments: increasing excitation frequency and material density cause increasing attenuation.

DETERMINATION OF THE PARAMETERS OF CANCELLOUS BONE USING LOW FREQUENCY ACOUSTIC MEASUREMENTS

The Biot model is widely used to model poroelastic media. Several authors have studied its applicability to cancellous bone. In this article the feasibility of determining the Biot parameters of cancellous bone by acoustic interrogation using frequencies in the 5–15 kHz range is studied. It is found that the porosity of the specimen can be determined with a high degree of accuracy. The degree to which other parameters can be determined accurately depends upon porosity.

A method of ascent for solving boundary value problems

Stefan Bergman [ l ] and Ilya Vekua [4] have given representation formulas for solutions of the partial differential equation (1). We obtain an improvement of their results for the case of two independent variables (namely equation (2) with n set equal to 2). Furthermore, we are able to extend our result to higher dimensions (the ascent) by a remarkably simple variation of this two dimensional formula. Our representation (2) also contains Vekuas formulas [4, p. 59], for the Helmholtz equation in n è 2 variables.

Determination of the parameters of cancellous bone using high frequency acoustic measurements

The Biot model is widely used to model poroelastic media. Several authors have tested its applicability to cancellous bone, but to do so requires a priori estimation of the parameters of the Biot model, which is an uncertain and expensive endeavor. A method of computing acoustic pressure in the low 100 kHz range is developed.

Acoustic response of a rigid-frame porous medium plate with a periodic set of inclusions.

The acoustic response of a rigid-frame porous plate with a periodic set of inclusions is investigated by a multipole method. The acoustic properties, in particular, the absorption, of such a structure are then derived and studied. Numerical results together with a modal analysis show that the addition of a periodic set of high-contrast inclusions leads to the excitation of the modes of the plate and to a large increase in the acoustic absorption.

Function theoretic solutions to problems of orthotropic elasticity

The plane strain problem for a two dimensional orthotropic elastic body is investigated. In particular analytic representations for the solution of the displacement boundary value problem and the stress boundary value problems are found. To this end, the Navier equations are reduced by means of composite transformations to normal form. These are the so-called equations for bianalytic function of the type (λk). The generalized Cauchy integral formula for this function theory is used to obtain representation formulae. A simplified method to solve these problems by bianalytic function theory is given for certain situations of plane strain for an orthotropic elastic body. AMS (MOS): 35A20, 35CO5, 35G15, 35J55.

The unidentified object problem in a shallow ocean

This work addresses the inverse problem of the identification of a passive three-dimensional impenetrable object in a shallow-water environment. The latter is assumed to have flat perfectly reflecting (sound-soft top and sound-hard bottom) boundaries and therefore acts as a guide for acoustic waves. These waves are employed to interrogate the object and the scattered acoustic wavefield is measured on the surface of a (virtual) vertically oriented cylinder (of finite or infinite radius, corresponding to near- or far-field measurements) fully enclosing the object. The direct scattering problem is resolved in approximate manner by employing, in a local manner, the known separated-variable solution for a scattering by a vertically oriented cylinder in a perfect waveguide. The inverse problem is resolved in the same manner (i.e., with the same approximate field ansatz) by least-squares matching of theoretical fields (for trial objects) to the measured field. Examples are given of successful shape reconstructio...

On Riemann boundary value problems for certain linear elliptic systems in the plane

The partial differential equations which occur in the theory of elastic plates and shells are among those which may be reduced to a first order elliptic system of the type studied by Douglis [3]. Under certain regularity conditions for the coefficients, a Beltrami transformation exists takmg the general first-order, elliptic systems into a normal-Douglis-form. This form can be further simplified, and more concisely represented by utilizing the algebra CY of hypercomplex numbers. The theory of solutions to these linear systems is known as generakzed hyperunai’yticfimction theory (see Gilbert [S, 6, 71 and Gilbert and HiIe [g, 9])* and bears the same relationship to the Douglis theory of hyperanalytic functions as Vekua’s theory to analytic functions. In [I] Hilbert boundary value problems for generalized hyperanalytic functions were studied. Subsequently semilinear Douglis systems were treated by Wendland [13] and Gilbert [7j. Th e p resent work deals with Riemann boundary value problems for linear systems. It is clear that our results may be extended to nonlinear hypercomplex systems resembling the complex cases investigated by Warowna-Dorau [14] and Wolska-Bochenek 1151. A good survey of the methods encountered in the analytic case may be found in the monographs of Gakhov [4J and Muskhelishvili [lo]. The fundamental kernels for the linear system (see [S]) permit the formulation of the Riemann boundary value problem for generalized hyperanalytic functions as a Cauchy type integral relation. In general, there is no similarity principle

Bergman's Integral Operator Method in Generalized Axially Symmetric Potential Theory

This paper contains a study of properties of solutions to the equation of generalized axially symmetric potentials. These potentials play an important role in many aspects of mathematical physics, in particular to an understanding of compressible flow in the transonic region. The ideas that have been basic in this investigation are contained in the integral operator method of Bergman. This method allows one to transplant certain properties of analytic functions to the solutions of linear partial differential equations. Results are obtained concerning singularities, residues, bounds, and growth of entire solutions, which are analogous to those found in classical function theory.