Wave equation

The long-time behavior of the solution for x >> t^1/3 of
the Korteweg-de Vries equation is found when the initial
data consists of a right or left-truncated soliton. The
initial data in either case is found to evolve into a
complete soliton of smaller amplitude. The amplitude,
velocity, and phase shift of the resultant soliton is explicitly
given, and the emergence of this soliton from
the initial disturbance is described in both cases.

Sound propagation in a waveguide is greatly dependent on the acoustic properties of the boundaries. The effect of these properties can be described by a bottom reflection coefficient RB, and surface reflection coefficient RS. Two methods for estimating reflection coefficients are used in this research. The first, the ratio method, is based on the variations of the Green's function with depth utilizing the ratio of the wavenumber spectra at two depths. The second, the pole method, is based on the wavenumbers of the modal peaks in the spectrum at a particular depth. A method to invert for sound speed and density is also examined. Estimates of RB and RS based on synthetic data by the ratio method were very close to their predicted values, especially for higher frequencies and longer apertures. The pole method returned less precise estimates though with longer apertures, the estimates were better. Using experimental data, results of the pole method as well a geoacoustic inversion technique based on them were mixed. The ratio method was used to estimate RS based on the actual data and returned results close to the predicted phase of p.

This thesis investigates inversion techniques used to determine the geoacoustic properties of a shallow-water waveguide. The data used were obtained in the Shallow Water '06 Modal Mapping Experiment in which four buoys drifted over a system of subbottom channels. The method used was perturbative inversion using modal eigenvalues as input parameters, which were found using an autoregressive spectral estimator. This work investigates the differences between a "channel" region and a "no channel" region based on an inferred stratigraphic model. Inversions were performed on data from a single buoy both at individual frequencies and multiple frequencies simultaneously. Since the use of multiple frequencies and a certain set of constraints proved to be an effective method of inversion, the method was applied to data from the other three buoys as well. It is shown that the "channel" and "no channel" regions have significantly different sound speed profiles.