A Simple Introduction to Water Waves

The theory of the water waves is the main subject in coastal hydrodynamics and plays a significant role in applied mathematics and in physics. In these notes we present the basics of the water wave theory. Specifically, after introducing briefly the basic concepts of continuum mechanics, we derive the physical laws describing the physics of an inviscid, incompressible fluid, namely the Euler equations. Euler equations are the governing equations of water waves, but because of the great difficulties in the theoretical and numerical studies of these equations, simple, approximate mathematical models have been derived instead. The models are usually simplified so as to be valid for specific types of waves.Water waves are usually divided into categories depending on their amplitude and their wavelength compared to the water depth. Some waves such as the Tsunami waves are generated in the deep ocean as waves of small amplitude with large wavelength, but as they approach the shoreline they grow in amplitude while their wavelength is decreased. Here, we will derive models for long waves with or without using the small amplitude assumption.