In this chapter, we derive the balance laws by requiring that the material volume, i.e., a volume that comprises a fixed set of particles, obeys the axioms of mass conservation, balance of linear and angular momentum, and the laws of thermodynamics. The important point to note is that the choice of this material volume is arbitrary; for example, any arbitrary subset of a given material volume also constitutes a valid material volume to which the above axioms can be applied. The original statements of all the axioms are in integral form. Using the arbitrariness of the material volume yields a set of differential equations governing the field variables. The solution of these differential equations subject to appropriate boundary and initial conditions then defines the variation of each field variable with space and time.

In fluid mechanics, it is useful to derive balance laws for a control volume, which is a region of space where various flow quantities are observed. The control volume approach has the advantage that, under certain conditions, it is possible to calculate quantities such as the force exerted or the power generated or dissipated, simply by having the appropriate information at the control surface. However, the weakness of this approach is that we do not obtain the details of the various fields at every point within the control volume. For obtaining detailed information, we need to solve the governing differential equations subject to appropriate boundary and initial conditions.

As emphasized in Chapter 1, we write the governing equations in tensorial form, because then it is immediately evident that such equations are independent of the choice of coordinate system. Only while solving specific problems, we choose a particular coordinate system, and write the tensor equations in component form in that particular coordinate system.

In this chapter, we study the problem of wave formation in an incompressible fluid. Waves can be of two types. The first type are surface gravity waves that are the ones seen, say, on the surface of an ocean. The second type are the ones in which the particles move to and fro in the direction of wave propagation. Such waves are known as compression or pressure waves. In this chapter, we deal only with surface waves occurring in an incompressible fluid. The waves occurring in a compressible fluid are dealt with in Chapter 8.

The assumption of the fluid being inviscid is found to be effective for formulating the problem of wave formation in an incompressible fluid. In addition, the waves are assumed to originate from a fluid that is originally at rest, and hence irrotational. Then Kelvin's theorem guarantees that the subsequent flow remains irrotational. Thus, effectively, we assume potential flow. Though this is the same assumption as made in most part of Chapter 3, the techniques for analyzing wave phenomena are different from those used previously.

In this chapter, we deal with both small amplitude surface waves and large amplitude shallow water (or ‘long’) waves. The equations governing wave motion are nonlinear. However, small amplitude waves can be treated by linearizing the governing equations. In the case of shallow water waves, however, we use the method of characteristics for solving the nonlinear partial differential equations. We start by presenting the governing equations for surface waves.

Governing Equations for Surface Waves

Consider a body of fluid with a top free surface, and the bottom surface bounded by a solid boundary, as shown in Fig. 4.1. Waves exist at the top free surface of the fluid. The x-axis is fixed at the mean level of the free surface, which is defined by the equation y = η(x, z, t). The bottom surface need not be a flat surface, but it is assumed to be invariant with respect to time.