434 SURFACE WAVES. [CHAP. IX 



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 



= A sinh -^ sinh -^ cos kas . cos (&amp;lt;rf+e) ......... (xiv), 



V^ V* 



with a2 = 



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 



&amp;lt;f&amp;gt; = A (sinh ay sin fiz + sin /3y sinh az) cos ksc . cos (crt+f) ...... (xvi). 



This leads in the same manner as before to 



a 2 -/3 2 = 2 ................................. (xvii), 



o- 2 sinh ah = ga cosh ah,} 

 and ..................... (xvm &amp;gt; 



whence ah coth ah ^h cot fth ........................... (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 (&amp;lt;rt+e) ........................... (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= *j3z, provided 



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



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



^7 (A cosh kh + B sinh kh) = A sinh kh + B cosh M, } 



9* [ ...... (xxiii). 



2&amp;lt;r 2 / , ,kh D . , M\ 4 . , kh D , kh ( 



-j- ( A cosh -zr-B sinh ) = A sinh -5- / cosh \ 



gk \ 2.2/ L * ) 



* &quot;Waves in Canals,&quot; Proc. Lond. Math. Soc., t. xxv., p. 101 (1894). 



