408 SURFACE WAVES. [CHAP. IX 



The pressure-formula is 



D 



- = const. gy + kc 2 (a cosh ky + /3 sinh ky) cos kx ... (4), 

 approximately, and therefore along the stream-line -\Jr = 

 - = const, -f (kc z a g(3) cos kx, 



so that the condition for a free surface gives 



kc*OL-g@ = ........................... (5). 



The equations (3) and (5) determine a and j3. The profile of the 

 free surface is then given by 

 y = /3 cos kx 



= T-TT - 77-^ u / 1 cos&# ............ (6). 



cosh &n $r/A;c 2 . Sinn &/& 



If the velocity of the stream be less than that of waves in still 

 water of uniform depth h, of the same length as the corrugations, 

 as determined by Art. 218 (4), the denominator is negative, so 

 that the undulations of the free surface are inverted relatively to 

 those of the bed. In the opposite case, the undulations of the 

 surface follow those of the bed, but with a different vertical scale. 

 When c has precisely the value given by Art. 218 (4), the solution 

 fails, as we should expect, through the vanishing of the denomi 

 nator. To obtain an intelligible result in this case we should be 

 compelled to take special account of dissipative forces. 



The above solution may be generalized, by Fourier s Theorem, so as to 

 apply to the case where the inequalities of the bed follow any arbitrary law. 

 Thus, if the profile of the bed be given by 



^- I dk( 



J * &amp;gt;o 



that of the free surface will be obtained by superposition of terms of the type 

 6) due to the various elements of the Fourier-integral ; thus 



1 r ji r 

 =-\ dk \ 

 TT J J _ 



/(A)cos(#-A) 



y=- u/ //9 -T_/i 



TT _ 00 cosh kh - kc* . smh kh 



In the case of a single isolated inequality at the point of the bed verti 

 cally beneath the origin, this reduces to 



_ ^ 



cosh kh - kc* . sinh kh 



/; 



r ~ 9-- u 



u cosh u-ghjc*. smh u 



,(iii), 



