28 



The total kinetic and potential energy per wave length is 



£^=-g-(^l-4.93^j (33) 



Of this amount of energy, one-half is transmitted forward with the 

 wave form and, therefore, the power transmitted per miit of crest 

 width is 



Shallow-water waves. — The orbits of these waves are (16) "ellipses 

 instead of circles, the eccentricity of the ellipses depending upon the 

 ratio of the wave length to the depth of water." 



For a given wave length, the eccentricity decreases with an increase of depth, 

 until at depths greater than half a wave length the ellipses are scarcely distinguish- 

 able from circles, while in very shallow water they tend to approach right lines. 



The orbits decrease in size below the surface, their focal distance remaining 

 constant, the vertical axes therefore decreasing more rapidly relatively than the 

 horizontal, until, at the bottom, were the* latter horizontal and frictionless, the 

 vertical axes would become zero, and the particles would move in horizontal 

 straight lines of length equal to the focal distance of the elliptical orbits of the 

 upper particles. 



Each particle revolves about the center of the ellipse with an angular velocity 

 which is not constant, but is greater in the vicinity of the crest and trough, and 

 less at midheight than if the orbit were a circle. 



These statements of Gaillard have been quoted in full because they 

 described in words, the behaviour mathematically described by equa- 

 tions 4a, 46, 5, and 6. The equation for the ratio of semi-minor to semi- 

 major axes of the elliptical orbits is: 



b. ^ , 2Td 

 -=tanh-j-- 

 a, L 



._ e^-l (6) 



47rd 



e ^ + 1 



The equations describing the reduced surface are: 



x=Rd—asmd (35a) 



y=h cose (356) 



The origin is taken as the orbit center of a crest particle and the 

 surface point fixed as shown in figure 1 1 . The wave velocity is given 

 by: 



C^J^^9^=J5.12^L (8a) 



\ as2Tr \ a, ^ ^ 



The total kinetic and potential energy per wave length is: 



£=^'(l- 19.74^)' (36) 



