Advanced Fluid Mechanics

Many different mechanisms for wave propagation in fluids exist. Compressibility of both liquids and gases allows for compression waves such as sound waves and shock waves. These generally have quite large velocities of propagation. Normally, they cannot be seen by the human eye without suitable instrumentation or lighting. Waves on surfaces and interfaces, on the other hand, are generally slower and can be observed easily. Gravity, inertial forces, surface tension, and viscosity are also important mechanisms in generating waves.
Much of the work presented in this chapter was originated by three English scientists: Lord Kelvin (born William Thomson, 1824 1907), Lord Rayleigh (born William Strutt, 1842 1919), and Sir Horace Lamb (1848 1934). The names in the literature for the first two can be confusing, because, for example, both Kelvin and Thomson are used, depending on whether the date of publication preceded or followed the bestowing of honors. Much of the work is summarized in the books by Rayleigh (1945) and Lamb (1932), and in the lengthy review in Wehausen and Laitone (1960).
Consider the case of small amplitude waves in a two-dimensional channel of constant depth h, an infinite length in the x direction, and having a uniform stream of velocity U. It is convenient to choose a coordinate system with an origin on the undisturbed free surface. When dealing with an unsteady flow problem with a free surface, it is necessary to find the pressure by using the...