In estuaries, the part of a river where the flow is predominantly driven by the tide, the water motion is important to understand. For instance, by understanding the flow, one can better predict how pollutants are transported, or when flooding is likely to occur. However, much is still unknown about certain aspects of the water motion, such as how the hydrodynamics are influenced by the steepness of the river bed.

In this thesis, the effect of steep lateral inclines on the river bed is systematically investigated. Because these steep inclines come with significant mathematical challenges, a new model for the water motion in estuaries is developed.

To study the effects of steepness, the model is applied to short estuaries with increasingly deep channels with increasingly steep inclines in the middle of the river, and shallower waters closer to the banks. The results indicate that during flood, i.e. the moment when flow into the estuary is strongest, water flows in through the deep channel and then from the channel towards the banks. During ebb, this pattern is reversed. For estuaries with steeper beds, currents are stronger, and the previously mentioned pattern is more pronounced.

Through a process known as momentum advection, so-called residual currents appear that are present regardless of the phase of the tidal cycle. Because these residual currents produce net transport by the movement of water, they are crucial in many transport processes involving sand, mud, salt, pollutants or other materials. The structure of the current is as follows: in the deeper channels, the residual current flows from the sea into the channel, and close to the banks,water flows out of the estuary. Steeper beds cause the residual flow to become stronger and (indirectly) because of the Coriolis effect, notably asymmetric.

This thesis is about the wave equation. The wave equation describes waves that propagate through a certain medium. The solution of the wave equation is a mathematical description of what that wave looks like. There are many fields of study where the wave equation is used to model processes that behave like travelling waves. For instance, a vibrating string or membrane can be modelled by the wave equation very well. Furthermore, the wave equation can be used to model light or sound waves and how they reflect and refract when travelling
through different materials.
Because the wave equation is such an important tool to model these phenomena, there are many people working on solving the wave equation in different contexts, i.e. finding a solution. However, it is difficult (often impossible) to find an exact solution. Therefore, people usually calculate a ’solution’ that is approximately correct.
An important question to ask is: ’Does the wave equation always have a solution?And is there only one solution?’. Physically, the answer is clear. If a string is vibrating, it clearly cannot vibrate in two ways at the same time and it will always vibrate in some particular way. Even though the answer is clear physically, answering this question mathematically is more difficult. It is good to remember that the wave equation is just a mathematical model of propagating waves, and it could very well be that the wave equation has more than one solution or no solution at all in some particular context.
The answer of this question is important for the people calculating approximate solutions to the wave equation. If there does not exist a unique solution, the process of calculating the approximate solution may break down.
Answering that question is the subject of this thesis. We will first introduce the necessary mathematical tools, after which we will use existing research to find out under which conditions the wave equation has one, and only one solution. Finally, we will extend our results to more general equations than just the wave equation.