## Dissertations, Theses, and Capstone Projects

9-2015

Dissertation

Ph.D.

Physics

Leon Cohen

Physics

#### Keywords

Modes; Noise; Non-Stationary; phase space; Snell's Law; Wigner

#### Abstract

This thesis consists of research regarding pulse and noise propagation in dispersive media. The research consists of three parts. In Part I we develop an approach for the propagation of non-stationary noise in waveguides, and in particular, we focus on the two-plate waveguide, which is a standard model for the ocean. he fundamental aim is to obtain the propagation of the space-time autocorrelation function. In our formulation, the noise is described by a Wigner spectrum, from which the autocorrelation function can be obtained. We discuss how to obtain the Wigner spectrum of a noise field from the Wigner spectrum of the noise that created it. We show that the Wigner spectrum of the field can be expressed in terms of the Wigner spectrum of the driving noise function and the Wigner distribution of the Green's function. We calculate the Green's function and its Wigner distribution for the two-plate waveguide. A number of special cases are considered, and we show how our result reduces to a known special case.

In Part II, we study the evolution of non-stationary noise in dispersive media in terms of modes and show how modes evolve and how they are effected by sources. Each mode satisfies a Schrödinger-type equation where the `"Hamiltonian'' may not be Hermitian. The Hamiltonian operator corresponds to the dispersion relation where the wavenumber is replaced by the wavenumber operator. A complex dispersion relation corresponds to a non-Hermitian operator and indicates that we have attenuation. The case of arbitrary initial conditions is developed. The results expand on a previous approximation in terms of the Wigner distribution of a single mode. This solution requires one to obtain the initial modal functions from the given initial wave, and the cross-Wigner approximation between different modal functions. The approximation is obtained for general linear wave equations by transforming the equations to phase space, and then solving in the new domain. A number of examples are given.

In Part III we consider the motion of a "particle'' in a medium of variable index of refraction whose motion is governed by Snell's law. A stratified medium is considered. We give a derivation of the Newtonian forces that govern the motion and show that a position-dependent variable mass is necessary. Explicit expressions are given for the velocity and acceleration components of the particle. These are derived directly from Snell's law. It is further shown that momentum is conserved along the interface, and that Snell's law follows from this conservation law. We apply the equations of motion and find the conditions for a SOFAR channel, a phenomenon where rays are trapped in a channel.

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