THEORY OF SIMPLE WAVES 



325 



fc2 = {2w/\y- = w-[(m\'L-) + (nV'Tr^)] (78) 



where m and /; are whole numbers 1, 2, 3, . . ., L is the 

 tank length and W the width. If L > W, the component 

 oscillation of the longest period is obtained by making 

 m = 1, ?!- = 0; the wave motion is then parallel to the 

 long side L of the rectangle and the wave length X = 2L. 

 The crest is at one end of the tank simultaneously with 

 the trough at the other end. The wave component or 

 mode corresponding to m = 2, i.e., X = L, is often also 

 of interest. In this case crests occur simultaneously at 

 both ends with the trough in the middle, or vice versa. 

 Generallv all modes can occur sinuiltaiieouslv, and the 



wave elevation is represented by the double Fourier's 

 series. 



r; = ilw o„,.„ cos {mT.x/L) COS imvz/W) (79) 



and the velocity potential of each inilividual mode is 



ga cosh k{y + h) 



0) 



</> 



COS im-wx/L) COS {n-wz/W) (80) 



cosh hh 



In writing these eciuations the origin of co-ordinates is 

 taken at f)ne corner of the tank, so that the tank is 

 bounded by the vertical planes at .r = and L, z = 

 and IF, with y negative downwards, and h the depth of 

 the tank. 



-Nomenclature- 



k. 



a = wave mnpliluilc (more e.\:ii-tly, 



half the height) 

 A = a coefficient 



c = wave celerity 

 E — wave energy 

 El. = kinetic wave energy 

 E^ = potential wave energy 



h = water depth 



k = 27r/X. Wave number 



(/ = acceleration of gravity 

 k^ = coefficients of accession to inertia 

 in ilirections of x and y-axes 



n = an index 



/) = pressure 

 r = a [lolar co-ordinate 

 / = time 

 T = wave period 

 V, v = horizontal and vertical cimipo- 

 ncnts of orliital water velocity 

 ZJ = water velocity 

 w = complex potential 

 a = included half-angle at wave cus)) 

 a. ji = horizontal and vertical semi- 

 axes of water-particle path in 

 water of limited depth 



rj = horizontal and vertical displace- 

 ments of water particles in 

 orbital motion 



1} = wave surface elevation 



6 = a polar co-ordinate 



d = maximum wave slope 



X = wave length 



H — ooellicient of viscosity 



V = kinematic viscosity, yi/p 



p = mass density 



4> = velocity potential 



\p = stream function 



w = circular freciuency = 'Iir /T 



