With a large number of waves (a large n), energy decreases with 

 increasing m, and the leading wave will eventually lose its identity. 

 At the group center, energy increases and decreases rapidly - to nearly 

 maximum and to nearly zero. Consequently, an energy front is located 

 at the center wave group for deepwater conditions. If waves had been 

 examined for shallow rather than deep water, the energy front would 

 have been found at the leading edge of the group. For any depth, the 

 ratio of group to phase velocity (C„/C) generally defines the energy 

 front. Also, wave energy is transported in the direction of phase propa- 

 gation, but moves with the group velocity rather than phase velocity. 



2,239 Summary - Linear Wave Theory . Equations describing water surface 

 profile particle velocities, particle accelerations, and particle displace- 

 ments for linear (Airy) theory are summarized in Figure 2-6. 



2.24 HIGHER ORDER WAVE THEORIES 



Solution of the hydroynamic equations for gravity-wave phenomena can 

 be improved. Each extension of the theories usually produces better 

 agreement between theoretical and observed wave behavior. The extended 

 theories can explain phenomena such as mass transport that cannot be 

 explained by linear theory. If amplitude and period are known precisely, 

 the extended theories can provide more accurate estimates of such derived 

 quantities as the velocity and pressure fields due to waves than can 

 linear theory. In shallow water, the maximum wave height is determined 

 by depth, and can be estimated without wave records. 



When concern is primarily with the oscillating character of waves, 

 estimates of amplitude and period must be determined from empirical data. 

 In such problems, the uncertainty about the accurate wave height and 

 period leads to a greater uncertainty about the ultimate answer than does 

 neglecting the effect of nonlinear processes. Thus it is unlikely that 

 the extra work involved in using nonlinear theories is justified. 



The engineer must define regions where various wave theories are 

 valid. Since investigators differ on the limiting conditions for the 

 several theories, some overlap must be permitted in defining the regions. 

 Le Mehaute (1969) presented Figure 2-7 to illustrate approximate limits 

 of validity for several wave theories. Theories discussed here are indi- 

 cated as are Stokes' third- and fourth-order theories. Dean (1973), 

 after considering three analytic theories, presents a slightly different 

 analysis. Dean (1973) and Le Mehaute (1969) agree in recommending cnoidal 

 theory for shallow-water waves of low steepness, and Stokes' higher order 

 theories for steep waves in deep water, but differ in regions assigned 

 to Airy theory. Dean indicates that tabulated stream function theory is 

 most internally consistent over most of the domain considered. For the 

 limit of low steepness waves in transitional and deep water, the differ- 

 ence between stream function theory and Airy theory is small. Additional 

 wave theories not presented in Figure 2-7 may also be useful in studying 



2-33 



