9.1 Introduction

In the present chapter we analyze closer the important case where the characteristic depth h ₀ is much smaller than the characteristic horizontal length ? in the wave motion i.e. ? = ?/ h ₀ ? 1). Though ? is not always equal to the wave length these waves are usually called "long waves". At first we will particularly examine the case where the dimensionless wave amplitude ? = A/ h ₀ = O( ? ²)) so that the Ursell parameter HL ²/ h ³ is O(1). This is also meaningful because we found in Chapter 7 that the equations for the other two cases ( ? ? ? ²) and ( ? ? ? ²) can actually be deduced from this general case simply by omitting the proper terms in the basic equations.

For clarity, we look at the simplest case of waves moving in one horizontal direction only (chosen as the x-axis) and on a constant depth h. This is defendable since all the important mechanisms will still be present. We will also continue to consider the equations in dimensionless form. The advantage of working with the dimensionless version is that the magnitudes of terms extracted in Chapter 7 remain explicit which greatly helps guiding the derivations. However, to simplify the equations we omit the ' that indicated dimensionless variables in Chapter 7. In order to keep contact...