Analytical and computational study of linear and nonlinear acoustics in quasi-1-D and 2-D gaseous and biphasic media
Date of Award
Doctor of Philosophy (PhD)
Mechanical and Aerospace Engineering
Sarma L. Rani
Sivaguru S. Ravindran
Sound waves, Helmholtz equation
The overall goal of this dissertation is to perform analytical and computational investigations of acoustic wave propagation in gaseous and biphasic domains with non-uniform cross-section and axially inhomogeneous mean properties. This research was carried out in five parts, with each successive part building on the previous one. The first part focuses on deriving the Helmholtz equation (or the wave equation in frequency space) governing the acoustic field in quasi one-dimensional (1-D) ducts. This derivation also introduced two linearly exact relations that were necessary to agree identically with the solutions from the Euler equation - the pressure-density relation and the linearly exact derivative boundary condition. The second part relates to the nonlinear dynamics of the time-dependant amplitudes of acoustic modes in a quasi 1-D duct, wherein the pressure fields obtained from the first part are used in conjunction with finite element method with the application of the standard Galerkin method. Two analytical methods were also investigated where the limit-cycle amplitude and frequency of pressure oscillations are quantified analytically using the Lindstedt--Poincaré perturbation method, and the transient evolution to the limit cycle is captured by the method of averaging. In the third part, the impact of non-isentropic effects on the acoustic field in a quasi 1-D duct is investigated and quantified. The non-isentropic effects were introduced via the heat conduction terms, and novel Wentzel--Kramers--Brillouin (WKB) methods were developed to obtain analytical solutions to the complex governing wave equations. The fourth parts investigates both spatial and temporal domains of acoustics in two-dimensional (2-D) non-uniform ducts, wherein we derived the wave equation in frequency space that governs the fluctuating pressure contours, as well as the nonlinear system of modally coupled equations governing the temporal amplitudes. The fifth part applies the fundamentals of acoustic wave modeling to the finite-element model (FEM)-based simulations of ultrasound propagation through the extracellular matrix (ECM) of the cartilage tissue. This ECM contained within itself multiple chondrons (chondrocytes embedded in a thin layer of pericellular matrix. Dilatation and displacement waves governed by Biot equations in porous biphasic media were solved by discretizing the domain into triangular elements with quadratic shape functions. The qualitative and quantitative insights gained from our study may be relevant to designing ultrasound-based therapies for osteoarthritis.
Basu, Sattik, "Analytical and computational study of linear and nonlinear acoustics in quasi-1-D and 2-D gaseous and biphasic media" (2023). Dissertations. 281.