Before we commence our presentation of the theory of water waves, we require a firm and precise base from which to start. This must be, at the very least, a statement of the relevant governing equations and boundary conditions. However, it is more satisfactory, we believe, to provide some background to these equations, albeit within the confines of an introductory and relatively brief chapter. The intention is therefore to present a derivation of the equations for inviscid fluid mechanics (Euler's equation and the equation of mass conservation) and a few of their properties. (The corresponding equations for a viscous fluid – primarily the Navier–Stokes equation – appear in Appendix A.) Coupled to these general equations is the set of boundary (and initial) conditions which select the water-wave problem from all other possible solutions of the equations. Of particular importance, as we shall see, are the conditions that define and describe the surface of the fluid; these include the kinematic condition and the rôles of pressure and surface tension. Some rather general consequences of coupling the equations and boundary conditions will also be mentioned.

Once we have available the complete prescription of the water-wave problem, based on a particular model (such as for inviscid flow), we may ‘normalise’ in any manner that is appropriate.