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 propagation, but moves with the group velocity rather than phase 

 velocity. 



i. Summary of 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. 



4. Higher Order Wave Theories. 



The solution of the hydrodynamic 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 the lin- 

 ear theory. If the precise measurements for amplitude and period are known, 

 the extended theories will provide estimates of such derived quantities as the 

 velocity and pressure fields due to waves that are more accurate than that 

 provided by linear theory. In shallow water, the maximum possible wave height 

 is determined by the depth and can be estimated without wave records. 



When concern is primarily with the oscillating character of waves, esti- 

 mates of amplitude and period must be determined from empirical data. In such 

 problems, the uncertainty of the accurate wave height and period leads to a 

 greater uncertainty of the ultimate answer than does neglecting the effect of 

 nonlinear processes. Therefore, 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 theo- 

 ries, some overlap must be permitted in defining the regions. Le Mehaute 

 (1969) presented Figure 2-7 to illustrate the approximate limits of validity 

 for several wave theories. Theories discussed here are indicated as the 

 Stokes third- and fourth-order theories. Dean (1974), after considering three 

 analytic theories, presents a slightly different analysis. Dean (1974) 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 difference between stream-function theory and Airy theory 

 is small. Other wave theories may also be useful in studying wave phenomena. 

 For given values of H, d, and T, Figure 2-7 may be used as a guide in 

 selecting an appropriate theory. The magnitude of the Ursell or Stokes para- 

 meter U„ shown in the figure may be used to establish the boundaries of 

 regions where a particular wave theory should be used. The parameter was 

 first noted by Stokes (1847) when he stated that the parameter must be small 

 if his equations were to remain valid for long waves. The parameter is 

 defined by 



l2h 

 U = -^ (2-45) 



R d^ 



2-31 



