Development of rigorous methods in fluid mechanics and theory of water waves

Development of rigorous methods in fluid mechanics and theory of water waves

The thesis consists of five chapters where the following problems are considered - the transformation of long linear waves in an ocean with a variable depth; long wave scattering in a canal with a rapidly varying cross-section; long linear surface waves on stationary currents in a canal of constant depth and variable width; and a resonant interaction of a solitary wave with external pulse-type perturbations within the framework of forced Korteweg-de Vries equation.

Chapter 1 reviews the history of these problems, and notes previous literature and research in this area.

In Chapter 2, the transformation of long linear waves in an ocean of a variable depth is studied. The transformation coeffcients are considered as functions of frequency and the total depth drop for three typical models of bottom profile variation: piecewise linear, piecewise quadratic and hyperbolic tangent profiles. The results obtained for all these profiles are analysed and compared from the physical point of view.

In Chapter 3, long wave scattering in a canal with a rapidly varying cross-section is studied. The scattering coeffcients are calculated for all possible incident wave orientations (background current downstream and upstream with respect to the background flow); and current types subcritical, transcritical, and supercritical. It is shown that when the over-reflection or over-transmission occurs, negative energy waves can appear in the flow. A spontaneous wave generation can happen in a transcritical accelerating flow, resembling a spontaneous wave generation on the horizon of an evaporating black hole due to the Hawking effect in astrophysics.

In Chapter 4, long wave transformation is studied in the canal with abrupt variation of width and depth. Again, all possible wave orientation with respect to the background current is considered, and all types of a current is studied (subcritical, transcritical, and supercritical). The transformation coefficients are calculated and analysed as functions of wave frequency, Froud number, and depth drop.

In Chapter 5, we revise the solutions of the forced Korteweg{de Vries equation for the resonant interaction of a solitary wave with the various external pulse-type perturbations. In contrast to the previous studies, we consider an arbitrary relationship between the width of a solitary wave and external forcing.

In many cases, exact solutions of the forced Korteweg{de Vries equation can be obtained for the specific forcings of arbitrary amplitude. The theoretical outcomes obtained by asymptotic method are in good agreement with the results of direct numerical modelling within the framework of forced Korteweg-de Vries equation.

In Conclusion, we summarise the results obtained within the various models and equations; discuss the future applications and innovation of the results; and suggest further perspectives for the future research.