Theory of Short and Long Waves 



27 



equations of continuity motion (see vol. I). If the .v-axis is horizontal in the 

 free surface at rest and if the positive r-axis is vertical downwards, and 



q = \, we have 



d 2 x dx IcPz _ \ dz_ ( p 

 'a¥da + \dr 1 g 8a + da 



0, 



<Px dx ld*z _ \ dz dp _ 

 dt 2 dy + \dt 2 8 )dy + ~dy~ ' 



(11.18) 



D = 



dx dz dx dz 



dD 

 dt 







< a ( y dy da ' 



a and y are the co-ordinates of this particle at rest at true /. Figure 14 

 shows that in general 



x = a + rsin# , z = y -+- rcos& , 



Fig. 14. Orbit of a water particle B (a and 7 are the co-ordinates of its centre). 



and if & = (xx—at), the path of any particle will be a circle with a radius r, 

 which can be a function of the depth y. After a time T = 2x/o and at a distance 

 ;. = 27i/x, this circular motion is repeated. If such a motion is possible, then 

 the equations (11.18) must be satisfied. The expression for D gives 



dr 



D = 1 + xr — + 



dy \dy 



dr 



wncostf 



As Z> should be independent of t, 



dr 



dy' 



= — xr , r = Ae~ y -y 



(11.19) 



The radii of the circular orbits decrease with depth with an ^-power. 

 As e~ nls = 201, we have the simple rule established by Rankine, according 

 to which, with each increase of depth of one-ninth of the wave length, the 

 amplitude is reduced to half of the previous amount. Accordingly, 



