Date of Award

Summer 2001

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Gregory A. Kriegsmann

Second Advisor

Jonathan H.C. Luke

Third Advisor

David C. Stickler

Fourth Advisor

Peter G. Petropoulos

Fifth Advisor

Edip Niver

Abstract

Some of the most challenging problems in acoustics and electromagnetics involve the study of scattered fields in waveguides caused by targets of elaborate shape. The complexity of the resulting scattered field depends on the geometry of the scatterer, and exact solutions exist only for the simple geometries.

The asymptotic methods developed in this dissertation give the approximate solutions for the scattered fields in two practically important geometries: an object placed inside a stratified waveguide, and a waveguide with multiple abrupt width transitions. The solutions for these geometries are obtained by approximating the field near the target or junction by the field that would be present if the waveguide walls were removed. This approximation is shown to be accurate if the waveguide width is large enough.

An asymptotic solution for the direct problem of scattering from the small object inside a stratified waveguide is obtained and investigated. Based on that approximation the solution to the problem of target localization for low frequency fields, or obstacles with certain symmetries, is developed. The solution of the shape reconstruction problem is demonstrated for an object in the high frequency limit.

An asymptotic solution for the direct problem of field scattering in the waveguide with multiple abrupt width transitions is also developed. Both Dirichlet and Neumann boundary conditions are considered. The applicability conditions of this approximation are investigated and the resulting accuracy is discussed. Numerical simulations supporting the validity of these asymptotic approximations are presented in each case.

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Mathematics Commons

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