Integrating by means of the substitution 



t-i i-f 



1/2 



{a-l)t 

 t-1 



1/2 



[108b] 



2 = K In 



1 +T 

 1 -T 



1 



In 



\ra- T 



yfa yfa + T 



[108c] 



Here it can be assumed that the amplitudes oi t, t - 1, t - a, and hence also of (;; - a)./{t - 1) 

 range only from to n, inclusively, those ofr , 1 + t and \/~a + t from to tt/2, and those of 

 1 - T and \ra - t from -77 to 0. 



As ^ -* - oo, T -♦ 1; hence the real part of z becomes infinite. Thus i = =» at /^ and 

 < = - oo at ^, since t increases from A to B. Thus, on the ^-plane the streamline AB becomes 

 the negative real axis, the line CDEF, the positive real axis. Since the flow is toward BC, 

 the transformed flow on the ^-plane must be one of convergence toward < = 0. The simplest 

 type of such convergent flow is that due to a line sink at i! = 0, for which the complex potential 

 may be written 



w = c\n t, [108d] 



from Section 40, where m = + zi// and c is a real constant. 



To fix the constants, take the a;-axis parallel to the channel. Then, to provide the 

 assumed inflow at AF , it is necessary that at AF w ^ Uz + constant or dw/dz -* U; see 

 Section 35. But 



dw dw jdz c I t - a^^ ^^ 

 dz dtl dt " KXt -1 



[lOSe] 



When 1^1 is large, the last fraction becomes unity. Hence it is necessary that 



— = i/. 

 K 



[108f] 



As BC or < = is approached, dw/dz must reduce similarly to A, V/h„. Hence, from 

 Equation [108e], 



K 



[108g] 



The fact that it is thus obviously possible to make the solution represent the assumed 

 flow in distant parts of the channel confirms the choice of m as a function of t. 



267 



