434 SURFACE WAVES. [CHAP. TX 



We may next consider the asymmetrical modes. The solution of this type 

 which is analogous to Kelland's was noticed by Greenhill (I. c.). It is 



^ sinh -7?; cos Tex . cos 

 \M 



with a 2 



When kh is small, this makes o- 2 =#/A, so that the 'speed' is very great 

 compared with that given by the theory of 'long' waves. The oscillation 

 is in fact mainly transversal, with a very gradual variation of phase as we 

 pass along the canal. The middle line of the surface is of course nodal. 



When kh is great, we get * edge- waves,' as before. 

 The remaining asymmetrical oscillations are given by 



= A (sinh ay sin fte + sin fry sinh az) cos kx . cos (crt -f e) ...... (xvi). 



This leads in the same manner as before to 



a 2 -j8 2 = 2 ................................. (xvii), 



o- 2 sinh ah qa cosh ah,] 

 and o sin &-# cos /} ..................... (XVU1 >> 



whence ah coth ah = $h cot fih ........................... (xix). 



There are an infinite number of solutions, with values of /3A in the third, 

 fifth, seventh, ... quadrants, giving 3, 5, 7, ... longitudinal nodes, one of which 

 is central. 



3. The case of a canal with plane sides inclined at 60 to the vertical has 

 been recently treated by Macdonald*. He has discovered a very compre- 

 hensive type, which may be verified as follows. 



The assumption 



= p cos kx . cos (<rt+c) ........................... (xx), 



where 



P = A cosh kz+B sinh /b+cosh - ccosh +D sinh .. 



evidently satisfies the equation of continuity; and it is easily shewn that 

 it makes 



for y= \/3z, provided 



C=2A, D=-2B (xxii). 



The surface-condition, Art. 239 (4), is then satisfied, provided 



-^ (A cosh kh + B sinh kh} = A sinh A + B cosh M, 1 



* ( (xxiii). 



2<r 2 / , , kh D . , M\ , . , M D , M f 

 ^ ( A cosh -n~-B sinh -^- ) = ^ sm h ~%~B cosh 



* "Waves in Canals," Proc. Land. Math. Soc., t. xxv., p. 101 (1894). 



