11.1 Introduction and equations

The main developments in this chapter relate to linearized surface waves in water, but acoustic and electromagnetic waves will also be mentioned. We start from the wave equation (10.18), which was developed from the equations of momentum balance and mass conservation in shallow water. The wave elevation, ?, is small in comparison with the water depth, H. If the problem is periodic, we can write the wave elevation, ?, quite generally as

where ? is the angular frequency and may be complex. Equation (10.18) now becomes

or, for constant depth, H,

where the wavenumber, k = ?/ ? gH, is related to the wave length, ?, by k = 2 ?/ ?. The wave speed is c = ?/ k. Equation (11.3) is the Helmholtz equation [which was also derived in Chapter 10, in a slightly different form, as Eq. (10.18)] which models very many wave problems. This is only one form of the equation of surface waves, for which there is a very extensive literature. ^[1], ^[2], ^[3], ^[4] From now on all problems will be taken to be periodic, and the overbar on ? will be dropped. The Helmholtz equation (11.3) also describes periodic acoustic waves. The wavenumber k is now given by ?/ c, whereas in surface waves ? is the angular frequency and c is the...