The aim of this thesis is to derive two distinct model equations for shallow water waves in a continuously stratified fluid and with variable bottom topography. This thesis is divided into two parts. First, we derive a KdV-type shallow water model equation for bi-directional shallow water waves along the equator with a continuous depth-dependent density. Secondly, we derive a KdV-type shallow water model equation for unidirectional shallow water waves along the equator with a continuous depth-dependent density and a bottom that may vary in the direction of wave propagation. We derive both model equations from the governing equations using asymptotic expansion. We obtain model equations that describe the horizontal velocity component for each fixed depth. We derive exact solutions for the second model equation under the assumption that the bottom is slowly varying and perform an analysis of the effect of the varying bottom and the change of density on the waves.

Layman Summary
The goal of this paper is to study how waves move over a fluid in with an underlying current. Oceans displace debris and heat. Ocean models can be used to predict the movement of debris and even predict the effect of the movements of oceans on the climate. Modeling water can be improved by introducing more information from observations into the models. The goal of this paper is to combine the effect of varying density and an underlying current in a model for ocean water near the equator. Our model focuses on long, shallow waves. This allows us to simplify the model and solve it more easily. We combine methods from different sources to create a model for our problem. The model we create shows there are two types of solution waves: Periodic travelling waves and Solitons. Solitons are single waves that don’t change shape through time. We find that density influences the amplitude of waves and the perturbation of the current. We find that the underlying current influences the wave profile’s width as well as the amplitude.

Summary
Models of oceans can be used to predict the displacement of debris and even trace its path back to its origin. Oceans are a large influence on the weather and climate all over the world. Improving these models is therefore very useful.
There isn’t a general equation that describes all water dynamics. Even if there would be, we would not have a computer good enough to make all the necessary calculations for the model. Though there will not be a perfect model, there is still a lot of room to improve the current models. The accuracy of water models, specifically for oceans, can be improved in different ways. We can increase the resolution to model smaller, more intricate behavior. The model can also be improved by coupling more different phenomena.
The goal of this paper is to combine existing methods for modeling the propagation of waves in a model that describes the propagation of waves over a current in a stratified fluid. We assume the fluid to be inviscid and incompressible. We also assume there is no thermal conductivity. We choose to focus the model on long and shallow waves and changes in the current. These
assumptions mean we can choose to base the governing equations on the Euler equations for inviscid fluids. Through non-dimensionalisation and scaling transformations, we transform our model to unitless equations. We examine only the behavior of the leading order solutions by expanding the unknown variables as asymptotic series. The final equations that describe the wave propagation belong to the family of Korteweg-deVries equations. One solution to these equations is a sech2(θ) function. They represent soliton waves. These are
solitary waves that hold their shape through the combination of dispersion and the non-linear character of the waves.
We find that the density influences the amplitude of waves and the perturbation of the current because it shows up as a multiplicative factor. We find that the underlying current influences the wave profile’s width as well as the amplitude because it also shows up inside the θ term of the sech^2(θ) function.

The Rhine Region of Fresh Water Influence (ROFI) is a shallow frictional river plume in front of the Dutch coast. Each tidal cycle a new tidal plume front with fresh-water is released. Recently, internal gravity waves have been observed in this plume. Using a Froude number analysis, support for the internal wave generation mechanism by a tidal plume front is found. It is shown, that on averaged neap tides the density stratification is large. This results in a larger area where the internal waves can be released from the tidal plume fronts compared to spring tides. As the Rhine ROFI is located in shallow water, it is investigated what the effect of bathymetry variation is on internal waves. This is studied by extending the standard KdV model derivation to account for a variable bed. This resulted in a small correction on the propagation speed inside the KdV equation. Observations of internal waves have been used to validate the KdV model. By scaling analysis of the observed waves it is obtained that the relative wave height and relative depth balance for most of the observed events. For these events the wave period and velocity amplitude of the KdV model are well matched with the measurements. This showed that the KdV model may be used for a first estimate of internal waves in the Rhine ROFI. By developing a TGE fitting procedure, it was possible to obtain the pycnocline depth and the direction of wave propagation with only limited data available. Therefore, the parameters have been varied and the error between the TGE solution and the velocity potential obtained from the velocity measurements has been minimized.