Analysis and design of wave scattering by weakly non-uniform waveguides

Abstract

When waves propagate through a medium with small irregularities of the size of the order of many wavelengths, a number of interesting phenomena may happen such as wave localization and large sensitivity of the wave field behavior with respect to the medium irregularity variation. These phenomena occur due to the interaction of the incident wave field with the medium small irregularities, and open the possibility of designing the system irregularity to achieve a desired vibratory response. This has a variety of engineering applications, such as the the design of the sea bottom of coastal areas to provide protection against the incoming swells, or the design of the material and geometrical properties of the cross section of pipelines, risers and mooring lines such that vibration transmission is minimized. The objective of this thesis is to understand how to tune the medium small irregularity such that the interaction of the incident wave field with the medium irregularity generates a desired reflected wave field. A particular design problem of interest is the prediction of the minimum amount of changes in the medium irregularity needed to minimize wave transmission to a desired level for a given range of frequencies of interest. As a model problem, we considered disordered chains of repetitive systems with the size of the order of many wavelengths of the incident wave. We applied an asymptotic theory for wave propagation along the non-uniform chain. For weak coupling between subsystems, the asymptotic theory predicted new results, such as exponential small transmission due to wave tunneling and explained localization phenomena as a turning point problem. For strong coupling,

(cont.) the asymptotic theory provided fundamental understanding of the effects of the irregularity on wave propagation. Pipelines and risers can be modeled as slender beams under tensile force. To describe well the effects of small irregularity in beams vibration, we derived asymptotically a simpler governing equation for the vibration problem. This new equation is asymptotic with respect to the beam irregularity steepness, but under the restriction of constant product of the flexural rigidity by the mass per unit length and constant tensile force, this new equation is an exact equation for the beam vibration and has a Helmholtz-like form. Inverse scattering methods for the Helmholtz-like equations can be applied to design the beam non-uniformity such that desired wave scattering properties are achieved. We also constructed a high order asymptotic solution for the scattering of mono-chromatic waves by the irregularity in slender beams. The asymptotic method used is the WKB method, which is basically a wave refraction theory, but we improved it such that wave reflection and wave mode conversion were captured. The asymptotic approach developed in the previous problems is extended and applied to the interaction of linear surface gravity waves with a bottom topography which varies slowly with respect to the length scale of the incident wave field. The asymptotic theory captured wave reflection and transmission and wave mode conversion, which leads to a more complete asymptotic representation of the wave field ...