NORMAL MODE BENDS FOR CIRCULAR ELECTRIC WAVES 1297 



amplitude is 



'(^■' I = 2- I i 



„,fe) I = i ^ «"'" dz 



dz 



(7) 



This amplitude represents mode conversion loss and therefore has to 

 be kept as small as possible. 



In (7) the function ^(2), i.e., the taper function, is still undetermined. 

 Obviously it can be chosen so as to optimize the taper performance. A 

 taper of optimal design keeps the unwanted mode below a certain value 

 with as short a taper length as possible. From (7) the relation between this 

 optimizing problem and the problem of the transmission line taper of 

 optimal design is at once evident. The transmission line taper is a low 

 reflection transition between lines of different characteristic impedances. 

 To minimize the length of the transition for a specified maximal reflec- 

 tion, the characteristic impedance has to change along the transition 

 according to a function which is essentially the Fourier transform of a 

 Tschebj^scheff polynomial of infinite degree. The same procedure can be 

 applied here and it will result in a curvature taper of optimal design. 



We are, however, at present not as much interested in a transition of 

 optimal design as in a curvature taper which can easily be built. Suppose 

 we bend the pipe to a bending radius Ro which causes only elastic defor- 

 mation. We do this on a form of radius Ro , Fig. 1. The forces acting on 

 both ends of the pipe cause a torque and hence a curvature of the pipe 

 which increases approximately linearly from the pipe end (z = 0) to the 

 point of contact (z = Zi) between pipe and form: 



k = ko-. (8) 



The corresponding curve which the pipe forms along the taper is Cornu's 

 spiral. 



We shall evaluate (7) for a curvature as given by (8). In considering 

 the mode conversion we may neglect all heat losses, that is 7 = jp, etc. 

 With 



c = Co-, (9) 



Zl 



we get 



^ _ 2co 1 



dz A|82i . i4_£oV_' (10) 



