In the preceding chapters, the discharge and the surface elevation were treated as continuous functions of s and t. This led to a coupled system of partial differential equations in two unknowns, with wave-like solutions. Certain flow systems can be schematized in terms of separate but connected components of finite dimension, in each of which we disregard the spatial variations. In each component, either storage or transport is modelled, but not both, so that the motion within them is not wave-like. In these cases, we speak of a discrete model. One such model is presented in this chapter.

Introduction

The disregard of spatial variations in the system components considered is allowed if the dimensions (l) are small compared with a typical length of the (long) waves in the domain. Stated another way, system components for which the travel time (‘residence time’) of long waves through them is short compared with the wave period. In such cases, phase differences within each of the system components are negligible. In other words, within each component the motion loses its wave-like character.

A good example of this category of situations is the tidal motion in a harbour basin. The water level in the basin can to a good approximation be assumed to be horizontal at all times, varying in time only. Its variation can be modelled with an ordinary differential equation instead of a partial differential equation, which simplifies the mathematics greatly.

The discrete modelling approach is utilized in the present chapter. It is relevant in itself, because numerous situations occurring in practice lend themselves to this approximation, and it is at the same time a preparation for the theory in the following chapter (on harmonic wave propagation) with respect to the linearized modelling of the flow resistance, and the use of complex algebra in the solution process. The advantage is that these building blocks, needed in the theory of Chapter 7, are introduced in the simpler context of the present chapter.

We focus on flow systems consisting of a nearly closed basin or reservoir, connected through some narrow, short opening or a channel of some length to an external body of water with a time-varying water level.