Thesis (M.S. in E.S.M.)--School of Civil and Environmental Engineering, Georgia Institute of Technology, 2004. Directed by Laurence Jacobs and Jianmin Qu. / Includes bibliographical references (leaves 113-117).

This thesis investigates the problems of applying Statistical Energy Analysis (SEA) tomodels that include solid volumes. Three wave types (Rayleigh waves, Pressure wavesand Shear waves) are important to SEA and the mathematics behind them is explainedhere. The transmission coefficients between the wave types are needed for energytransfer in SEA analysis and different approaches to solving the properties of wavepropagation on a solid volume are discussed. For one of the propagation problems, asolution, found in Momoi [6] is discussed, while the other problem remains unsolveddue to the analytical difficulties involved.

We consider one-dimensional longitudinal vibrations of a rigid rod with a nonuniform cross-section, fixed at its ends with lumped masses and springs. The cross-section inertia effects are taken into account on the basis of the Rayleigh theory. The equation of motion and the boundary conditions are derived from Hamilton’s variational principle. The characteristic equation is constructed and the eigenvalues for the harmonic vibrations of the rod are calculated. It is shown that the eigenvalues are bounded from above. Two types of the orthogonality of the eigenfunctions corresponding to the eigenvalues are discussed. The Green function is constructed for the problem of forced vibrations of the rod governed by a linear fourth-order partial differential equation, which involves mixed derivatives. Exact solutions of the rod vibration problems are found for rods with constant and conical cross-sections. Rigid isotropic waveguides are often used for generating, transmitting, and amplifying mechanical vibrations, for example, in acoustic transducers. Theoretical investigation of acoustic, mechanical, and electromagnetic waveguides is usually based on the analysis of second-order wave equations. This approach is justified in descriptions of the wave propagation in relatively thin and long rigid rods. As was shown by Rayleigh [1], the error due to the neglect of the transverse motion of the rod is proportional to the square of the ratio of the characteristic section radius to the length of the rod (aspect ratio). For a more accurate analysis of the longitudinal vibrations of a relatively thick and short rod, the rod deformation in the transverse direction must also be taken into account. The approach to the analysis of the vibrations of a thick and short rod used in this study is based on the theory of longitudinal vibrations of a rod, in which the effects due to the transverse motion are taken into account (the corresponding mathematical model is called the Rayleigh rod). The equation of motion and the boundary conditions for the onedimensional longitudinal vibrations of the Rayleigh rod with variable cross section and ends fixed by means of lumped masses and springs are derived from Hamilton’s variational principle. As a result, we arrive at a linear fourth-order partial differential equation with variable coefficients, which involves mixed derivatives. Previously, approximate analytical methods, such as the Galerkin method [2] and the method based on the expansion of the solution in a power series in the Poisson coefficient [3], were used for solving this equation. The frequencies of the natural vibrations of a cylindrical rod with rigidly fixed ends were determined in [4, pp. 159, 160]. In this study we use the method of the separation of variables based on the exact solutions of the equations of motion of the Rayleigh rod, which makes it possible to construct the Green function. A similar approach to an analysis of the longitudinal vibrations of stepped rigid waveguides described by second-order wave equations was applied in [5, 6].

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This dissertation descibes the development and the application of a planar acoustic microscope lens. The prototype lens consists of a pair of concentric circular metal electrodes plated on the interface between a piezoelectric solid and a liquid medium. These two circular electrodes excite Rayleigh waves of velocity V(,R) which are converging towards the common centre but which are phase matched to the compressional waves of the velocity V(,c) in the liquid in a very narrow range of zenith angle about a value given by (phi)(,m) = sin('-1)V(,c)/V(,R). The waves radiated into the liquid are thus in the form of a hollow cone converging onto a common focal spot on the lens axis at a distance determined by this zenith angle and the radius of the electrodes. This planar acoustic microscope lens is called the Rayleigh-to-Compressional Conversion (RCC) lens. / Since the lens behavior is determined by the geometry of the electrodes and because of the simplicity of the photolithographic fabrication process of the RCC lens, more complicated configurations can be made as easily as the prototype; for example, semicircular lenses have been produced and analyzed. / A mathematical analysis based on a spatial impulse of stress applied on the solid/liquid boundary has been used to calculate the focussing characteristics of the RCC lens. For isotropic solid not only the particle displacements of the compressional wave in the liquid have been computed, but also that of the waves radiated into the solid. For anisotropic solids only the radiation pattern of the compressional wave in the liquid, which is the one of most interesting, has been investigated using an isotropic equivalent model. In the model the circular shape of the electrodes has been considered to consist of many line segments and it has been used to analyze the focal properties of partial circles and anisotropic substrate. / This planar acoustic microscope lens has been employed in standard transmission and reflection imaging experiments to demonstrate the structure of its focal spot and in particular the lack of spherical aberration when traversing a metal surface. Because of the hollow conical nature of the beam away from the focal region the RCC lens is inherently adapted to dark field microscopy. Some properties of semicircular lens are also given as examples in linear and nonlinear operation. . . . (Author's abstract exceeds stipulated maximum length. Discontinued here with permission of author.) UMI

This dissertation addresses forward and inverse modeling of Rayleigh waves for near surface investigation. Results were obtained by imaging abandoned mine openings using Rayleigh waves in the laterally inhomogeneous medium. The efficient staggered grid stencil method to solve elastic wave equations using 2-D finite difference technique is presented. This numerical scheme is used to conduct a series of parametric studies on the propagation of Rayleigh waves. The first parametric study was conducted on a flat layered model of increasing and decreasing velocity with depth. A Rayleigh waves dispersion curve is found to be sensitive on a layer’s depth up to half of the minimum wavelength of Rayleigh waves. The phase velocity in the dispersion curve of Rayleigh waves is inversely and directly proportional to the frequency, depending on velocity increase or decrease with depth. The parametric study was carried out by introducing dipping layers in the model with increasing dip. The front (near the shot point) and back (at the end of receiver line) shot records are different if the subsurface contains dip. Dispersion is observed in near offset for down dip and in the far offset for up dip, computed from front and back shots respectively. Finally, a parametric study looked at subsurface anomalies with different shapes and sizes as well as their material properties. A Rayleigh wave is sensitive to very high material contrast and very low material contrast of the anomaly from it surrounding medium. The presence of a low material contrast anomaly from the surrounding medium traps the energy which causes reverberation. A Rayleigh wave is sensitive to an anomaly which is placed within the depth between one-third to half of minimum wavelength of Rayleigh wave from the surface. In order to resolve lateral heterogeneity, a new method is developed in this research which allows localization of the multichannel record in different panels. The dispersion curve of Rayleigh waves is computed in each panel using the slant stack technique. On the basis of parametric studies, an innovative inversion algorithm has been developed to minimize the error norm; ”the sum of the squares of the difference of reference and model dispersion curves” in an iterative way using a Very Fast Simulated Re-annealing (VFSR) technique.

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Thesis (Ph.D.) -- University of Adelaide, Dept. of Physics, 1967. / Typescript.