240] SPECIAL CASES. 435 



* 



The former of these is equivalent to 



A=H\ coshM - ^ysinhM ), 



V gk ) 



B H\^-r cosh kh sinh kh 

 and the latter then leads to 



+ l=0 ..................... (xxv). 



Also, substituting from (xxii) and (xxiv) in (xxi), we find 



|+ A^L.(xxvi). 



The equations (xxv) and (xxvi) are those arrived at by Macdonald, by a 

 different process. The surface- value of P is 



\) 

 H ...... (xxvn). 



--- ^ 



The equation (xxv) is a quadratic in <r 2 /gk. In the case of a wave whose 

 length (2?r//0 is great compared with A, we have 



3M 2 



nearly, and the roots of (xxv) are then 



\kh, and <P\gk=\lkh .................. (xxviii), 



approximately. If we put o- = kc, the former result gives c i = \gh, in accord- 

 ance with the usual theory of ' long ' waves (Arts. 166, 167). The formula 

 (xxvii) now makes P=3II, approximately; this is independent of y, so that 

 the wave ridges are nearly straight. The second of the roots (xxviii) makes 

 (r 2 =g/h, giving a much greater wave-velocity ; but the considerations adduced 

 above shew that there is nothing paradoxical in this*. It will be found on 

 examination that the cross-sections of the waves are parabolic in form, and 

 that there are two nodal lines parallel to the length of the canal. The period 

 is, in fact, almost exactly that of the symmetrical transverse oscillation 

 discussed in Art. 237, 2. 



When, on the other hand, the wave-length is short compared with the 

 transverse dimensions of the canal, kh is large, and coth |M = 1, nearly. The 

 roots of (xxv) are then 



a*lgk=\ and <r*\gk=\ ........................ (xxix), 



approximately. The former result makes P=H, nearly, so that the wave- 

 ridges are straight, experiencing only a slight change of altitude towards the 



* There is some divergence here, and elsewhere in the text, from the views 

 maintained by Greenhill and Macdonald in the papers cited. 



282 



