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The nonlinear interactions between long surface waves and interfacial waves in a two-layer fluid are studied theoretically. The fluid is density-stratified and the thicknesses of the top and bottom layers are both assumed to be shallow relative to the length of a typical surface wave and interfacial wave, respectively. A set of Boussinesq-type equations are derived for potential flow in this system. The equations are then analyzed for the dynamics of the nonlinear resonant interactions between a monochromatic surface wave and two oblique interfacial waves. The analysis uses a second order perturbation approach. Consequently, a set of coupled transient evolution equations of wave amplitudes is derived. Moreover, the effect of weak viscosity of the lower layer is incorporated in the problem and the influences of important parameters on surface and interfacial wave evolution (namely the directional angle of interfacial waves, density ratio of the layers, thickness of the fluid layers, surface wave frequency, surface wave amplitude, and lower layer viscosity) are investigated. The results of the parametric study are discussed and are generally in qualitative agreement with previous studies. In shallow water, a triad formed of surface waves (or interfacial waves) can be considered in near-resonant interaction. In contrast to the previous studies which limited the study to a triad (one surface wave and two interfacial waves or one interfacial and two surface waves), the problem is generalized by considering the nonlinear interactions between a triad of surface waves and three oblique pairs of interfacial waves. In this system, each surface wave is in near-resonance interaction with other surface waves and in exact resonance with a pair of oblique interfacial waves. Similarly, each interfacial wave is in near-resonance interaction with other interfacial waves which are propagating in the same direction. Inclusion of all the interactions considerably changes the pattern of evolution of waves and highlights the necessity of accounting for several wave harmonics. Effects of density ratio, depth ratio, and surface wave frequency on the evolution of waves are discussed. Finally, a formulation is derived for spatial evolution of one surface wave spectrum in nonlinear interaction with two oblique interfacial wave spectra. The two-layer Boussinesq-type equations are treated in frequency domain to study the nonlinear interactions of time-harmonic waves. Based on weakly two-dimensional propagation of each wave train, a parabolic approximation is applied to derive the formulation.

The depth-averaged Saint-Venant equations, which are used for most computational flow models, are adequate in simulating open channel flows with insignificant curvatures of streamlines. However, these equations are insufficient when applied to flow problems where the effects of non-hydrostatic pressure distribution are predominant. This study provides a comprehensive examination of the feasibility of a simple one-dimensional Boussinesq-type model equation for such types of flow problems. This equation, which allows for curvature of the free surface and a non-hydrostatic pressure distribution, is derived using the momentum principle together with the assumption of a constant centrifugal term at a vertical section. Besides, two Boussinesq-type model equations that incorporate different degrees of corrections for the effects of the curvature of the streamline are investigated in this work. One model, the weakly curved flow equation model, is the simplified version of the flow model based on a constant centrifugal term for flow situations that involve weak streamline curvature and slope, and the other, the Boussinesq-type momentum equation linear model is developed based on the assumption of a linear variation of centrifugal term with depth.

The depth-averaged Saint-Venant equations, which are used for most computational flow models, are adequate in simulating open channel flows with insignificant curvatures of streamlines. However, these equations are insufficient when applied to flow problems where the effects of non-hydrostatic pressure distribution are predominant. This study provides a comprehensive examination of the feasibility of a simple one-dimensional Boussinesq-type model equation for such types of flow problems. This equation, which allows for curvature of the free surface and a non-hydrostatic pressure distribution, is derived using the momentum principle together with the assumption of a constant centrifugal term at a vertical section. Besides, two Boussinesq-type model equations that incorporate different degrees of corrections for the effects of the curvature of the streamline are investigated in this work. One model, the weakly curved flow equation model, is the simplified version of the flow model based on a constant centrifugal term for flow situations that involve weak streamline curvature and slope, and the other, the Boussinesq-type momentum equation linear model is developed based on the assumption of a linear variation of centrifugal term with depth.

We prove that the solution of the Boussinesq equation with relatively small initial data exists globally and decays exponentially under some boundary conditions. / Ph. D.

The prediction of coastal groundwater movement is necessary in coastal management. However, the study in this field is still a great challenge due to the involvement of tidal-groundwater interactions and the phenomena of hydrodynamic dispersion between salt-fresh water in the coastal region. To date, numerous theories for groundwater dynamic have been made available in analytical, numerical and also experimental forms. Nevertheless, most of them are based on the zeroth-order shallow flow, i.e. Boussinesq approximation. Two main components for coastal unconfined aquifer have been completed in this Thesis: the vertical beach model and the sloping beach model. Both solutions are solved in closed-form up to higher order with shallow water parameter ([epsilon]) and tidal amplitude parameter ([alpha]). The vertical beach solution contributes to the higher-order tidal fluctuations while the sloping beach model overcomes the shortcomings in the existing solutions. From this study, higher-order components are found to be significant especially for larger value of [alpha] and [epsilon]. Other parameters such as hydraulic conductivity (K) and the thickness of aquifer (D) also affect the water table fluctuations. The new sloping solution demonstrated the significant influence of beach slope ([beta]) on the water table fluctuations. A comprehensive comparison between previous solution and the present sloping solution have been performed mathematically and numerically and the present solution has been demonstrated to provide a better prediction

There exists substantial applications motivating the study of nonlinear longitudinal wave propagation in layered (or laminated) elastic waveguides, in particular within areas related to non-destructive testing, where there is a demand to understand, reinforce, and improve deformation properties of such structures. It has been shown [76] that long longitudinal waves in such structures can be accurately modelled by coupled regularised Boussinesq (cRB) equations, provided the bonding between layers is sufficiently soft. The work in this thesis firstly examines the initial-value problem (IVP) for the system of cRB equations in [76] on the infinite line, for localised or sufficiently rapidly decaying initial conditions. Using asymptotic multiple-scales expansions, a nonsecular weakly nonlinear solution of the IVP is constructed, up to the accuracy of the problem formulation. The asymptotic theory is supported with numerical simulations of the cRB equations. The weakly nonlinear solution for the equivalent IVP for a single regularised Boussinesq equation is then constructed; constituting an extension of the classical d'Alembert's formula for the leading order wave equation. The initial conditions are also extended to allow one to separately specify an O(1) and O(ε) part. Large classes of solutions are derived and several particular examples are explicitly analysed with numerical simulations. The weakly nonlinear solution is then improved by considering the IVP for a single regularised Boussinesq-type equation, in order to further develop the higher order terms in the solution. More specifically, it enables one to now correctly specify the higher order term's time dependence. Numerical simulations of the IVP are compared with several examples to justify the improvement of the solution. Finally an asymptotic procedure is developed to describe the class of radiating solitary wave solutions which exist as solutions to cRB equations under particular regimes of the parameters. The validity of the analytical solution is examined with numerical simulations of the cRB equations. Numerical simulations throughout this work are derived and implemented via developments of several finite difference schemes and pseudo-spectral methods, explained in detail in the appendices.