fall. These waves may travel hundreds or thousands of kilometers after leav- 

 ing the area in which they were generated. Wave energy is dissipated inter- 

 nally within the fluid, by interaction with the air above, by turbulence on 

 breaking, and at the bottom in shallow depths. 



Waves that reach coastal regions expend a large part of their energy in 

 the near shore region. As the wave nears the shore, the wave energy may be 

 dissipated as heat through turbulent fluid motion induced by breaking and 

 through bottom friction and percolation. While the heat is of little concern 

 to the coastal engineer, breaking is important because it affects both beaches 

 and manmade shore structures. Thus, shore protection measures and coastal 

 structure designs are dependent on the ability to predict waveforms and fluid 

 motion beneath waves, and on the reliability of such predictions. Prediction 

 methods generally have been based on simple waves where elementary mathemati- 

 cal functions can be used to describe wave motion. For some situations, these 

 simple formulas provide reliable predictions of wave conditions; however, for 

 other situations the predictions may be unsatisfactory for engineering appli- 

 cations. Although many theoretical concepts have evolved in the past two cen- 

 turies for describing complex sea waves, complete agreement between theory and 

 observation is not always found. 



In general, actual water-wave phenomena are complex and difficult to 

 describe mathematically because of nonlinearities, three-dimensional charac- 

 teristics, and apparent random behavior. However, there are two classical 

 theories, one developed by Airy (1845) and the other by Stokes (1880), that 

 describe simple waves. The Airy and Stokes theories generally predict wave 

 behavior better where water depth relative to wavelength is not too small. 

 For shallow water, a cnoidal wave theory often provides an acceptable approxi- 

 mation of simple waves. For very shallow water near the breaker zone, sol- 

 itary wave theory satisfactorily predicts certain features of the wave 

 behavior. These theories are described according to their fundamental charac- 

 teristics, together with the mathematical equations that describe wave behav- 

 ior. Many other wave theories have been presented in the literature which, 

 for some specific situations, may predict wave behavior more satisfactorily 

 than the theories presented here. These other theories are not included, 

 since it is beyond the scope of this manual to cover all theories. 



The most elementary wave theory, referred to as small-amplitude or linear 

 wave theory, was developed by Airy (1845). This wave theory is of fundamental 

 importance since it is not only easy to apply, but also reliable over a large 

 segment of the whole wave regime. Mathematically, the Airy theory can be con- 

 sidered a first approximation of a complete theoretical description of wave 

 behavior. A more complete theoretical description of waves may be obtained as 

 the sum of an infinite number of successive approximations, where each addi- 

 tional term in the series is a correction to preceding terms. For some situ- 

 ations, waves are better described by these higher order theories, which are 

 usually referred to as finite-amplitude theories. The first finite-amplitude 

 theory, known as the trochoidal theory, was developed by Gerstner (1802). It 

 is so called because the free-surface or wave profile is a trochoid. This 

 theory is mentioned only because of its classical interest. It is not recom- 

 mended for application, since the water particle motion predicted is not that 

 observed in nature. The trochoidal theory does, however, predict wave pro- 

 files quite accurately. Stokes (1880) developed a finite-amplitude theory 

 that is more satisfactory than the trochodial theory. Only the second-order 



2-2 



