A comprehensive study of potential symmetries admitted by partial differential equations is given using the wave equation utt = c²(x)uxx as a given prototype equation, R. Methods are given for the construction of various conserved forms for R. Potential symmetries for R are nonlocal symmetries realized as local symmetries of auxiliary systems obtained from conserved forms of R. The existence of potential symmetries for R can be determined algorithmically and automatically by the use of a symbolic manipulation program. Most importantly, the potential symmetries obtained through one auxiliary system may or may not include and/or extend those obtained through another auxiliary system. The work in this thesis significantly extends the previously known classes of potential symmetries admitted by R and results in a better understanding of the limits in the construction of potential symmetries for R. / Science, Faculty of / Mathematics, Department of / Graduate

I present a new time-domain method for solving for the stress and particle velocity of normally incident plane waves propagating in a smoothly varying one-dimensional medium. Both the Young's modulus E and the density ⍴ are allowed to vary smoothly with depth. The restriction of geometrical optics, that the wavelength be much less than the material stratification length, is not required in this new method. The infinite geometrical optics expansion is truncated after n terms, imposing a condition on the acoustic impedance I for exact solutions to exist. For the case ռ = 2 there are three general classes of impedance functions for which the resultant expansion is uniform and exact. To check the numerical validity of the "truncated asymptotic" (TA) solution results are calculated for the case of a medium with a linear velocity gradient for which there is an exact solution in the frequency domain. Since a linear velocity gradient is not one of the foregoing classes of impedance functions, a curve-fitting approach is necessary. The TA method compares favourably to the exact solution and is accurate to within the error of the required curve fit. Two classes of synthetic seismograms are calculated for smooth velocity and density variations. The same impedance as a function of traveltime is used for both classes. In the first class the principal variation in impedance is due to velocity, while in the second it is mainly due to density. The amplitudes in both classes of synthetic seismograms are very similar, but, as expected, the traveltime curves for each class are widely separated. For the case ⍴ = constant the TA solution is used as a bench-mark to analyse a two-term WKBJ approximation for three classes of velocity functions. The velocity functions are such that the TA solutions are exact. For two of the classes the WKBJ solution performs well when the length of the transition zone is of the same order, or larger, than the length of the applied wavelet. For steeper velocity gradients the WKBJ solution begins to differ significantly from the exact TA solution. The WKBJ solution for the third class performs extremely well even for steep gradients. Equations governing the validity of the WKBJ solution are examined to explain the above results. Equations are derived to describe the distortion of a stress pulse propagating through a transition zone. For small velocity gradients (relative to the length of the applied pulse) the wavelet changes amplitude but its phase is not effected. As the gradient increases and the velocity function becomes a discontinuity at z = 0 the wavelet travels through undistorted. Only when the transition zone width is of the order of the length of the wavelet is there any visible phase distortion. Reflection and transmission coefficients as functions of time are calculated for low, intermediate and high gradient transition zones. The transmission coefficient is a delta function in each case. The reflection coefficient has the shape of a Hilbert transform for low gradients. For higher gradients the reflection coefficient approaches the shape of a delta function. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

by Chan Chi-kin. / Includes bibliographical references / Thesis (M.Ph.)--Chinese University of Hong Kong, 1985

Ho Chiu Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 100-103). / Text in English; abstracts in English and Chinese. / Ho Chiu Man. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Historical Background --- p.1 / Chapter 1.2 --- Motivations of the Project --- p.3 / Chapter 1.3 --- Outline of Thesis --- p.4 / Chapter 2 --- Review on a One-dimensional Vibrating Cavity --- p.5 / Chapter 2.1 --- The R Function --- p.5 / Chapter 2.2 --- Photon Generation --- p.7 / Chapter 2.3 --- Instantaneous Mode Expansion --- p.10 / Chapter 2.4 --- Vacuum Energy Density --- p.15 / Chapter 3 --- Graphical Method --- p.17 / Chapter 3.1 --- Construction of R-function --- p.17 / Chapter 3.2 --- Fixed-point Analysis --- p.19 / Chapter 3.3 --- A Special Class of Mirror Trajectories --- p.23 / Chapter 3.4 --- Further Analysis --- p.27 / Chapter 4 --- Wave Evolution in a One-dimensional Vibrating Cavity --- p.32 / Chapter 4.1 --- Instantaneous Mode Expansion Method --- p.32 / Chapter 4.2 --- Transformation Method --- p.34 / Chapter 4.3 --- R-Method --- p.34 / Chapter 4.4 --- Consistency between Different Methods --- p.35 / Chapter 5 --- Floquet's Theory --- p.40 / Chapter 5.1 --- System of Linear Differential Equations with Time-dependent Coefficients --- p.40 / Chapter 5.2 --- Fundamental Set of Solutions and Matrizant --- p.41 / Chapter 5.3 --- System of Linear Differential Equations with Periodic Coefficients --- p.42 / Chapter 5.4 --- Possible Properties of the Solution --- p.43 / Chapter 5.5 --- Eigenvalues for the One-dimensional Vibrating Cavity --- p.44 / Chapter 6 --- Photon Creation in an Oscillating Spherical Cavity --- p.46 / Chapter 6.1 --- Mode Decomposition --- p.46 / Chapter 6.2 --- Bogoliubov Transformation --- p.48 / Chapter 6.3 --- Photon Creation --- p.51 / Chapter 6.4 --- Mean Electric Field Operator --- p.53 / Chapter 6.5 --- Illustrations and Observations --- p.54 / Chapter 7 --- Multiple Scale Analysis --- p.61 / Chapter 7.1 --- Photon Creation by First-order Multiple Scale Analysis --- p.61 / Chapter 7.2 --- Parametric Resonance --- p.64 / Chapter 7.3 --- Nonstationary Medium --- p.71 / Chapter 7.4 --- Photon Statistics --- p.74 / Chapter 7.5 --- Finite Temperature Correction --- p.75 / Chapter 8 --- Squeezing Effect of the Classical Waves --- p.76 / Chapter 8.1 --- Squeezing Effect in the One-dimensional Vibrating Cavity --- p.76 / Chapter 8.2 --- Squeezing Effect in the Oscillating Spherical Cavity --- p.81 / Chapter 9 --- Supersymmetric Approach to Photon Generation --- p.88 / Chapter 9.1 --- Field Quantization --- p.88 / Chapter 9.2 --- Adiabatic Approximation --- p.90 / Chapter 9.3 --- Scattering Interpretation --- p.91 / Chapter 9.4 --- Supersymmetric Approach --- p.93 / Chapter 9.5 --- Examples --- p.94 / Chapter 10 --- Conclusion --- p.98 / Bibliography --- p.100

Cho, Chi Lam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 63-65). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

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Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (leaves [141]-148).

Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (leaves 87-88).