33 



C d 



Figure 13 shows the magnitude of the ratio -pf 8,s a function of y 



A 4:Trd , 11 • 1 47r</ 4:7rd J Co -, A 47r(Z 



As —jT- becomes very small, smh — =^=^r- and -^==1. As -j- 



Co 

 becomes large, -^ = 0.5. For waves in very shallow water, the wave 



velocity is the same for all wave lengths and the group velocity is the 

 same as the wave velocity. In deep water, the wave velocity depends 

 on the wave length (dispersive medium) and the group velocity is 

 one-half the wave velocity. The group velocity therefore falls in the 

 range 



0.5C<:Cg<C 



Probably the most important application of the group velocity lies 

 in the calculation of the rate of transmission of energy (18). Osborne 

 Keynolds (19) showed for deep-water waves of sinusoidal form that 

 one-half the energy was transmitted ahead. This result is in agree- 

 ment with the conclusions of the trochoidal theory. However, for 

 waves in any depth of water, Eayleigh (20) showed that the rate of 

 transmission of energy is 



^wd 

 wCaH ^ , L \=wCa^ 



L 



Now —^ — is the entire wave energy which advances with the group 



velocity. This result is said to be characteristic of all waves and pre- 

 sumably could be applied to wave forms and velocity-length relation- 

 ships found experimentally even though these results disagree with 

 theory. 



The general statement then appears to be that the power transmitted 

 per unit of crest width is 



P=^ (43) 



C«=C-if (44) 



If waves in nature are found to be very nearly trochoidal, Gaillard's 

 expression for the energy of both deep-water and shallow-water waves 

 may be used for E in equation 43. However, if experiment shows 

 the trochoidal equations to be not sufficiently accurate, the true value 

 is to be used in this equation, for the equation is general and does 

 not depend upon any particular wave theory. 



If experiment does not confirm equation 2 for the wave velocity 

 as a function of wave length, the true experimental relationship is 



dC 

 to be used in equation 44 for the group velocity. The derivative -rj 



sinh -f-/ 



