428 SURFACE WAVES. [CHAP. IX 



The next gravest mode is symmetrical, and is given by the lowest finite 

 root of (v), which is M=2'3650, whence o- = l'5244 (gjh^. 



In this mode, the profile of the surface has two nodes, whose positions are 

 determined by putting < = 0, z=k, in (ii); whence it is found that 



The next mode corresponds to the lowest finite root of (ix), and so ont. 



2. Greenhill, in the paper already cited, has investigated the symmetrical 

 oscillations of the water across a channel whose section consists of two straight 

 lines inclined at 60 to the vertical. In the (analytically) simplest mode of 

 this kind we have, omitting the time-factor, 



(t> + ty = iA(y + iz) 3 + B ........................... (xiv), 



or <t> = Az(z z -3t/*) + B, ^ = Ay(y*-W] ............... (xv), 



the latter formula making \^ = along the boundary y=x/3.z. The 

 surface-condition (iii) is satisfied for z=h, provided 



<r 2 =#/A, B = 2Ah? ........................... (xvi). 



The corresponding form of the free surface is 



a parabolic cylinder, with two nodes at distances of '5774 of the half-breadth 

 from the centre. Unfortunately, this is not the slowest mode, which must 

 evidently be of asymmetrical type. 



3. If in any of the above cases we transfer the origin to either edge of 

 the canal, and then make the breadth infinite, we get a system of standing 

 waves on a sea bounded by a sloping bank. This may be regarded as made 

 up of an incident and a reflected system. The reflection is complete, but 

 there is in general a change of phase. 



When the inclination of the bank is 45 the solution is 



<fr=H{ef cz (cos ky - sin ky} + e~ kv (cos kz + sin kz}} cos (a-t + e). . .(xviii). 

 For an inclination of 30 to the horizontal we have 



sin \k (y- 



) cosp(yW3*)}cos(<r* + ) ........ ...(xix). 



In each case a^glc^ as in the case of waves on an unlimited sheet of deep 

 water. 



These results, which may easily be verified ab initio, were given by 

 Kirchhoff (Z.c.). 



* Lord Rayleigh, Theory of Sound, Art. 178. 



f An experimental verification of the frequencies, and of the positions of the 

 loops (places of maximum vertical amplitude), in various fundamental modes, has 

 been made by Kirchhoff and Hansemann, " Ueber stehende Schwingungen des 

 Wassers," Wied. Ann., t. x. (1880); Kirchhoff, Ges. Abh., p. 442. 



