Inverse problem for the wave equation : partial data and novel boundary sources

An inverse problem for the wave equation is a mathematical formulation of the problem to convert measurements of sound waves to information about the wave speed governing the propagation of the waves. This doctoral thesis extends the theory on the inverse problems for the wave equation in cases with partial measurement data and also considers detection of discontinuous interfaces in the wave speed. A possible application of the theory is obstetric sonography in which ultrasound measurements are transformed into an image of the fetus in its mother's uterus. The wave speed inside the body can not be directly observed but sound waves can be produced outside the body and their echoes from the body can be recorded. The present work contains five research articles. In the first and the fifth articles we show that it is possible to determine the wave speed uniquely by using far apart sound sources and receivers. This extends a previously known result which requires the sound waves to be produced and recorded in the same place. Our result is motivated by a possible application to reflection seismology which seeks to create an image of the Earth s crust from recording of echoes stimulated for example by explosions. For this purpose, the receivers can not typically lie near the powerful sound sources. In the second article we present a sound source that allows us to recover many essential features of the wave speed from the echo produced by the source. Moreover, these features are known to determine the wave speed under certain geometric assumptions. Previously known results permitted the same features to be recovered only by sequential measurement of echoes produced by multiple different sources. The reduced number of measurements could increase the number possible applications of acoustic probing. In the third and fourth articles we develop an acoustic probing method to locate discontinuous interfaces in the wave speed. These interfaces typically correspond to interfaces between different materials and their locations are of interest in many applications. There are many previous approaches to this problem but none of them exploits sound sources varying freely in time. Our use of more variable sources could allow more robust implementation of the probing.

Analytical solution of the laminar mean flow wave equation in a lined or unlined two-dimensional rectangular duct

The prediction of the sound attenuation in lined ducts with sheared mean flow has been a topic of research for many years. This involves solving the sheared mean flow wave equation, satisfying the relevant boundary condition. As far as the authors' knowledge goes, this has always been done using numerical techniques. Here, an analytical solution is presented for the wave propagation in two-dimensional rectangular lined ducts with laminar mean flow. The effect of laminar mean flow is studied for both the downstream and the upstream wave propagation. The attenuation values predicted for the laminar mean flow case are compared with those for the case of uniform mean flow. Analytical expressions are derived for the transfer matrices.

Relaxation system based sub-grid scale modelling for large eddy simulation of Burgers' equation

An implicit sub-grid scale model for large eddy simulation is presented by utilising the concept of a relaxation system for one dimensional Burgers' equation in a novel way. The Burgers' equation is solved for three different unsteady flow situations by varying the ratio of relaxation parameter (epsilon) to time step. The coarse mesh results obtained with a relaxation scheme are compared with the filtered DNS solution of the same problem on a fine mesh using a fourth-order CWENO discretisation in space and third-order TVD Runge-Kutta discretisation in time. The numerical solutions obtained through the relaxation system have the same order of accuracy in space and time and they closely match with the filtered DNS solutions.

Exact analysis of Burgers equation on semiline with flux condition at the origin

The initial boundary value problem for the Burgers equation in the domain x greater-or-equal, slanted 0, t > 0 with flux boundary condition at x = 0 has been solved exactly. The behaviour of the solution as t tends to infinity is studied and the “asymptotic profile at infinity” is obtained. In addition, the uniqueness of the solution of the initial boundary value problem is proved and its inviscid limit as var epsilon → 0 is obtained.

Spectral approach for the soliton and periodic solutions of the nonlinear wave equation

A spectral method that obtains the soliton and periodic solutions to the nonlinear wave equation is presented. The results show that the nonlinear group velocity is a function of the frequency shift as well as of the soliton power. When the frequency shift is a function of time, a solution in terms of the Jacobian elliptic function is obtained. This solution is periodic in nature, and, to generate such an optical pulse train, one must simultaneously amplitude- and frequency-modulate the optical carrier. Finally, we extend the method to include the effect of self-steepening.

A formula for the solution of general Abel integral equation

A new formula for the solution of the general Abel Integral equation is derived, and an important special case is checked with the known result.

Exact N-Wave Solutions for the Non-Planar Burgers Equation

An exact representation of N-wave solutions for the non-planar Burgers equation u(t) + uu(x) + 1/2ju/t = 1/2deltau(xx), j = m/n, m < 2n, where m and n are positive integers with no common factors, is given. This solution is asymptotic to the inviscid solution for Absolute value of x < square-root (2Q0 t), where Q0 is a function of the initial lobe area, as lobe Reynolds number tends to infinity, and is also asymptotic to the old age linear solution, as t tends to infinity; the formulae for the lobe Reynolds numbers are shown to have the correct behaviour in these limits. The general results apply to all j = m/n, m < 2n, and are rather involved; explicit results are written out for j = 0, 1, 1/2, 1/3 and 1/4. The case of spherical symmetry j = 2 is found to be 'singular' and the general approach set forth here does not work; an alternative approach for this case gives the large time behaviour in two different time regimes. The results of this study are compared with those of Crighton & Scott (1979).

Generalised one-dimensional burgers'equation in relativistic fluiddynamics

This work deals with the effects of weak nonlinearity and weak dissipation on a linear wave in relativistic gasdynamics. Using perturbation and asymptotic expansions, a relativistic analogue of generalised one-dimensional Burgers' equation of classical gasdynamics is derived to describe far-field description of the wave. Steady state solution is presented for strict one-dimensional case.

On the solution of the equation ut+unux+H(x, t, u)=0

We consider the equation u(t) + u(n)u(x) + H(x, t, u) = 0 and derive a transformation relating it to u(t) + u(n)u(x) = 0. Special cases of the equation appearing in applications are discussed. Initial value problems and asymptotic behaviour of the solution are studied.

Applicability of Butler's equation in interpreting the thermodynamic behavior of surfaces and adsorption in Fe-S-O melts

Expressions for various second-order derivatives of surface tension with respect to composition at infinite dilution in terms of the interaction parameters of the surface and those of the bulk phases of dilute ternary melts have been presented. A method of deducing the parameters, which consists of repeated differentiation of Butler's equations with subsequent application of the appropriate boundary conditions, has been developed. The present investigation calculates the surface tension and adsorption functions of the Fe-S-O melts at 1873 and 1923 K using the modified form of Butler's equations and the derived values for the surface interaction parameters of the system. The calculated values are found to be in good agreement with those of the experimental data of the system. The present analysis indicates that the energetics of the surface phase are considerably different from those of the bulk phase. The present research investigates a critical compositional range beyond which the surface tension increases with temperature. The observed increase in adsorption of sulfur with consequent desorption of oxygen as a function of temperature above the critical compositional range has been ascribed to the increase of activity ratios of oxygen to sulfur in the surface relative to those in the bulk phase of the system.

Conjectures about phase turbulence in the complex Ginzburg-Landau equation

In the complex Ginzburg-Landau equation, we consider possible ''phase turbulent'' regimes, where asymptotic correlations are controlled by phase fluctuations rather than by topological defects. Conjecturing that the decay of such correlations is governed by the Kardar-Parisi-Zhang (KPZ) model of growing interfaces, we derive the following results: (1) A scaling ansatz implies that equal-time spatial correlations in 1d, 2d, and 3d decay like e(-Ax2 zeta), where A is a nonuniversal constant, and zeta=1/2 in 1d. (2) Temporal correlations decay as exp(-t(2 beta)h(t/L(z))), with the scaling law <(beta)over bar> = <(zeta)over bar>/z, where z = 3/2, 1.58..., and 1.66..., for d = 1,2, and 3 respectively. The scaling function h(y) approaches a constant as y --> 0, and behaves like y(2(beta-<(beta)over bar>)), for large y. If in 3d the associated KPZ model turns out to be in its weak-coupling (''smooth'') phase, then, instead of the above behavior, the CGLE exhibits rotating long-range order whose connected correlations decay like 1/x in space or 1/t(1/2) in time. (3) For system sizes, L, and times t respectively less than a crossover length, L(c), and time, t(c), correlations are governed by the free-field or Edwards-Wilkinson (EW) equation, rather than the KPZ model. In 1d, we find that L(c) is large: L(c) similar to 35,000; for L < L(c) we show numerical evidence for stretched exponential decay of temporal correlations with an exponent consistent with the EW value beta(EW)= 1/4.

Solution of a hypersingular integral equation of the second kind

A straightforward analysis involving the complex function-theoretic method is employed to determine the closed-form solution of a special hypersingular integral equation of the second kind, and its known solution is recovered.

An inverse problem for the Helmholtz equation involving two semi-infinite fluids

An analytical method is developed for solving an inverse problem for Helmholtz's equation associated with two semi-infinite incompressible fluids of different variable refractive indices, separated by a plane interface. The unknowns of the inverse problem are: (i) the refractive indices of the two fluids, (ii) the ratio of the densities of the two fluids, and (iii) the strength of an acoustic source assumed to be situated at the interface of the two fluids. These are determined from the pressure on the interface produced by the acoustic source. The effect of the surface tension force at the interface is taken into account in this paper. The application of the proposed analytical method to solve the inverse problem is also illustrated with several examples. In particular, exact solutions of two direct problems are first derived using standard classical methods which are then used in our proposed inverse method to recover the unknowns of the corresponding inverse problems. The results are found to be in excellent agreement.