454 SURFACE WAVES. [CHAP. IX 



When the velocity c of the stream is less than the minimum 

 wave- velocity, the factors of 



are imaginary. There is now no indeterminateness caused by 

 putting fju = ab initio. The surface-form is given by 



Q r coskx 



The integral might be transformed by the previous method, but it 

 is evident a priori that its value tends rapidly, with increasing x, 

 to zero, on account of the more and more rapid fluctuations in 

 sign of cos kx. The disturbance of level is now confined to the 

 neighbourhood of the origin. For x = we find 



Finally we have the critical case where c is exactly equal to 

 the minimum wave- velocity, and therefore K 2 = /q. The first term 

 in (10) or (11) is now infinite, whilst the remainder of the expres 

 sion, when evaluated, is finite. To get an intelligible result in 

 this case it is necessary to retain the frictional coefficient //. 



If we put // = 2sr2 ) we have 



(k-^ + i^ f ={k-(&amp;lt; + ^-iw}}{k-(&amp;lt;-w-\-iw}} ............ (vii), 



so that the integral (i) may now be equated to 



l-M r r &amp;lt;p* r j** \ . 



- \ I j -. -- ..dk- I -j r i - !~v&amp;lt;fifcV ...... (vm). 



4tzr \J 9 k-^-w+iw] J Q k-(K. + TZ-iw} J 



The formulae of Art. 227 shew that when or is small the most important 

 part of this expression, for points at a distance from the origin on either side, 



is 



It appears that the surface-elevation is now given by 



7T , 



/i/&amp;gt;\ 



(16). 



The examination of the effect of inequalities in the bed of a 

 stream, by the method of Art. 230, must be left to the reader. 



