The considerations and analyses in this chapter are based on a strongly reduced set of equations, viz. those for the modelling of low, long waves without resistance. We assume a horizontal open channel without longitudinal variations in the channel cross section. The channel does not have to be straight, but for simplicity we refer to it as a prismatic channel.We account for storage and (local) inertia while neglecting advective accelerations (consistent with the restriction to low waves) and resistance (restricting the validity of the results to rapid variations). Based on these simplifying assumptions, a partial differential equation will be derived, the wave equation, so called because it has wave-like solutions. We present the wave equation in its most elementary form: linearized, with constant coefficients, no resistance, no forcing.

Simple Wave

Propagation

The notion of wave propagation is crucial in the present context. In order to develop a good understanding, we approach it in three steps of increasing complexity: a qualitative description of the propagation of a so-called simple wave, travelling into a region of rest; a quantitative description of the same in terms of algebraic relations derived from overall balance equations for a finite control volume; and a description in terms of a partial differential equation, the so-called wave equation.

We first consider a positive translatory wave, as sketched in Figure 3.1a (see also Figure 2.1). We recognize three regions: an undisturbed region, a region of established uniform flow and a travelling transient or wave front between them.

Thanks to the slope of the free surface in the wave front, a pressure gradient exists there that causes an acceleration of the water particles in the direction from the high-water side to the low-water side, indicated in the figure with a double arrow. At a given location, the water is initially at rest, while it accelerates during the passage of the wave front. Once the front has passed that location, the local free surface is again horizontal, the pressure gradient is zero, and the flow velocity is constant (we ignore flow resistance).