Shallow Water Wave Transformation through External Factors 121 



water basin. Lord Rayleigh (1876, p. 260) has shown that this follows 

 from the law of conservation of energy. 



Sverdrup and Munk (1946) have presented this problem in a simple 

 way. In deep water, half of the total energy E travels at a velocity c 

 (see p. 22). In shallow water, only a fraction n of the energy E travels at 

 a velocity c. The fraction n is sometimes used to express the relationship 

 between the wave and group velocities. The ratio of the energies equals 



E_ 



1 Co 



In c 



(V.19) 



and the ratio of wave heights is proportional to the square root of the energy 



F.-1/lVfe?)- (v ' 20) 





As the wave period remains constant, and I = cT, the height H will be 

 approximately proportional to the fourth root of the depth. The ratio c/c 

 and n have both been expressed theoretically by Stokes as functions of 

 a ratio h/?. , where /? is the depth and / the wave length in deep water. This 

 ratio has been called "relative depth"; the exact derivation was given by 

 Munk and Traylor (1947). Figure 55 shows the theoretical relationship 



2-00 

 i-80 

 1-60 

 1-40 

 i-20 

 i-OO 

 0-80 



~~ 1 I ! ! I ! ! ! ! — 



Change in wave height before breaking 

 Berkeley wave far* 



0-01 



200 



1-80 



1-60 



1-40 



1-20 ^ 



100 



080 



0-02 



0-03 004 (0-062) 008 (0-108) (0 212) 3 0-4 0-6 0-8 10 



d/U 

 Fig. 55. Ratio wave height over deep-water wave height as function of relative depth. Line 

 gives theoretical relationship, filled-in figures show distribution of observed values. Theory 

 and observations both show initial decrease followed by increase (Sverdrup and Munk). 



between H/H and hjX , according to Sverdrup and Munk, from whom we 

 quote the following paragraphs/ 



"As the wave comes into shallow water, the theory shows first a decrease 

 of 8% in wave height and then an increase. The changes are small and 

 difficult to detect from individual observations. A statistical study of a large 

 number of observations made at the University of California in a wave tank 

 gives evidence of the dip and the subsequent increase. Field observations 

 have confirmed the computed increase, but have been too inaccurate to 



