#### Date of Award

Summer 1998

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Program/Concentration

Computational and Applied Mathematics

#### Committee Director

John Tweed

#### Committee Member

Mark H. Dunn

#### Committee Member

Gordon Melrose

#### Committee Member

F. Farassat

#### Committee Member

John H. Heinbockel

#### Abstract

The numerical solution of two classes of hypersingular integral equations is addressed. Both classes are integral equations of the first kind, and are hypersingular due to a kernel containing a Hadamard singularity. The convergence of a Galerkin method and a collocation method is discussed and computationally efficient algorithms are developed for each class of hypersingular integral equation.

Interest in these classes of hypersingular integral equations is due to their occurrence in many physical applications. In particular, investigations into the scattering of acoustic waves by moving objects and the study of dynamic Griffith crack problems has necessitated a computationally efficient technique for solving such equations.

Fracture mechanic studies are performed using the aforementioned techniques. We focus our studies on problems addressing the Stress Intensity Factors (SIF) of a finite Griffith crack scattering an out of plane shear wave. In addition, we consider the problem of determining the SIF of two parallel Griffith cracks and two perpendicular Griffith cracks. It is shown that the method is very accurate and computationally efficient.

In acoustics, we first consider the moving wing problem. For this problem we wish to find the sound produced by the interaction of a moving wing with a known incident sound source. Although this problem is relatively simple, it is a good precursor to the two-dimensional, finite, moving duct problem.

The bulk of the research is focused on solving the two-dimensional, finite, moving duct problem. Here we look at sound propagation and radiation from a finite, two-dimensional, moving duct with a variety of inlet configurations. In particular, we conduct studies on the redirection of sound by a so-called scarf inlet design. In said designs, we are able to demonstrate the ability to redirect sound away from sensitive areas.

#### DOI

10.25777/9v3k-n651

#### ISBN

9780599208841

#### Recommended Citation

St. John, Richard S..
"The Solution of Hypersingular Integral Equations With Applications in Acoustics and Fracture Mechanics"
(1998). Doctor of Philosophy (PhD), Dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/9v3k-n651

https://digitalcommons.odu.edu/mathstat_etds/110

#### Included in

Acoustics, Dynamics, and Controls Commons, Materials Science and Engineering Commons, Mathematics Commons