A general feature of wave phenomena is the transmission of information and energy through a physical system at a finite speed. A disturbance brought about somewhere in the system, e.g. due to operation of a control structure in an irrigation system, reaches other locations after a finite time. Insight into this phenomenon is important both for the purpose of effective control of water levels and discharges in the system and for performing the required computations. The so-called method of characteristics lends itself particularly well to this purpose because it makes visible how disturbances travel through the system and it enables their computation. It was developed by Massau (1878).

Introduction

In this chapter, we use the mass balance and the momentum balance without the low-wave approximations. Flow resistance is not included except for a minor reference.

As before, we restrict ourselves to one-dimensional systems, schematically represented by the s-axis, and consider the varying position of a disturbance in the course of time. This can be represented as a curve in the (s, t)-plane whose slope ds/dt equals the local propagation speed of the disturbance. Such curves are called characteristics. They portray how information travels through the system, as illustrated in Figure 4.10.

The balance equations for mass and momentum for one-dimensional wave phenomena form a set of two partial differential equations for two dependent variables, such as the depth (d) and the discharge (Q), as functions of two independent variables (s, t). The two dependent variables are called state variables. The instantaneous values of these can be represented as a point in the state plane, a plane with the two state variables as coordinates.

Given a set of sufficient initial and boundary conditions, the solution of the set of partial differential equations is determined. Expressed in terms of d and Q, this solution is a set of values d (s, t) and Q(s, t), which can be represented as a surface in the (d, s, t)-space and the (Q, s, t)-space, respectively, the so-called integral surfaces, depicted schematically in Figure 5.1.