52 Mr. J. M c Cowan on the Solitary Wave. 



and the volume of the wave elevation per unit breadth of 

 channel is 



v = \ v dx=2\ rjdx (28) 



J -oo Jo 



To perform the integration it is convenient to use the trans- 

 formation 



C™ sm n mhdx _ C h (cos m0— cos mh) n - 1 d6_ . , 9Q , 



Jo (cos mh + cosh mx) n J m sin n ~ l mh H £ j- v 



This gives, neglecting m^h 5 , 



A t = h, A 2 =hnK 2 -^m 3 h 4 , A 3 =^ 5 mVi 3 , . . (30) 

 hence 



^^^(l + lma + jLrf) .... (31) 



or, for a rough approximation, v = 2ah. 



Again, we see from (7) that the horizontal velocity in the 

 wave-motion of all particles in the same vertical line is 

 approximately the same, and that therefore each will be dis- 

 placed through nearly the same distance as the wave passes 

 over it. The average displacement 8 of the particles is easily 

 obtained, for as the wave passes completely across any section 

 of the channel it must convey over it a quantity of liquid 

 equal to its own volume, and this being done by the dis- 

 placement of the particles we must have v = hh, or 



S = v/h = 2a{l+Uia + j\-m 2 a 2 ). . . . (32) 



The displacement of the particles comes, however, most 

 naturally from the general discussion of the motion of the 

 particles, to which we proceed. 



7. The Motion of the Particles. 



So far it has been sufficient to consider the steady motion 

 resulting from the wave-motion by impressing on it a velocity 

 equal and opposite to that of the wave propagation, but it 

 now becomes necessary to consider the wave-motion itself. 



Let f and £ be the displacements parallel to the axes of x 

 and z respectively at time t, of the particle which was initially, 

 i. e. when t = — co , at x, z. Let yfr f and <j> r be the current 

 function and velocity potential respectively, at the point x -f- £, 

 2 + £ at time t, and for brevity put z f =z + £, x f =x + i;—TJt, 

 then by (4) and (5) 



■xJr' = JJa S * n mZ ' (33) 



cos mz' + cosh mx / ' * 



<£ = Va — t- ; j, . . . (34) 



cos mz' -j- cosn mx 7 ' 



