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 recommended for appli- 

 cation, since the water particle motion predicted is not that observed in 

 nature. The trochoidal theory does, however, predict wave profiles quite 

 accurately. Stokes (1880) developed a finite-amplitude theory which is 

 more satisfactory than the trochodial theory. Only the second-order Stokes' 

 equations will be presented, but the use of higher order approximations is 

 sometimes justified for the solution of practical problems. 



For shallow-water regions, cnoidal wave theory, originally developed 

 by Korteweg and De Vries (1895) , predicts rather well the waveform and 

 associated motions for some conditions. However, cnoidal wave theory has 

 received little attention with respect to actual application in the solu- 

 tion of engineering problems. This may be due to the difficulties in 

 making computations. Recently, the work involved in using cnoidal wave 

 theory has been substantially reduced by introduction of graphical and 

 tabular forms of functions. (Wiegel, 1960), (Masch and Wiegel, 1961.) 

 Application of the theory is still quite involved. At the limit of cnoidal 

 wave theory, certain aspects of wave behavior may be described satisfacto- 

 rily by solitary wave theory. Unlike cnoidal wave theory, the solitary 

 wave theory is easy to use since it reduces to functions which may be 

 evaluated without recourse to special tables. 



Development of individual wave theories is omitted, and only the 

 results are presented since the purpose is to present only that infor- 

 mation which may be useful for the solution of practical engineering 

 problems. Many publications are available such as Wiegel (1964), Kinsman 

 (1965) , and Ippen (1966a), which cover in detail the development of some of 

 the theories mentioned above as well as others. The mathematics used here 

 generally will be restricted to elementary arithmetic and algebraic opera- 

 tions. Emphasis is placed on selection of an appropriate theory in accord- 

 ance with its application and limitations. 



Numerous example problems are provided to illustrate the theory 

 involved and to provide some practice in using the appropriate equations 

 or graphical and tabular functions. Some of the sample computations give 

 more significant digits than are warranted for practical applications. 

 For instance, a wave height could be determined to be 10.243 feet for 

 certain conditions based on purely theoretical considerations. This 

 accuracy is unwarranted because of the uncertainty in the basic data used 

 and the assumption that the theory is representative of real waves. A 

 practical estimate of the wave height given above would be 10 feet. When 

 calculating real waves, the final answer should be rounded off. 



2.22 WAVE FUNDAMENTALS AND CLASSIFICATION OF WAVES 



Any adequate physical description of a water wave involves both its 

 surface form and the fluid motion beneath the wave. A wave which can be 

 described in simple mathematical terms is called a simple wave. Waves 



2-3 



