curvature k of the hub, which must be interpreted 

 as some mean value of the curvature of 6r (x) in 

 the wetted region not too far upstream. 



First we consider the amplitude of the waves on 

 the cavity. This amplitude is an increasing function 

 of both U and <, as follows from i) and iii) . Next 

 we consider the point of separation. An increase 

 of U or K tends to shift this point from the point 

 where 6ryj(x) = to the point where 6rh(x) = 5rh(-") 

 as follows from ii) and iv) . Hence in these respects, 

 an increase of U has roughly the same effect as an 

 increase of k. However, the length of the waves on 

 the cavity is, as follows from the previous theory 

 (87) , independent of k but is a decreasing function 

 of U. 



Finally we consider a shape of the hub which 

 induces, for some value of a, no waves on the cavity 

 at x = ". The existence of nontrivial shapes can 

 be shown as follows. Let 6rj^-[(x) and Sr^^j^'^^ tis 

 two shapes of the hub and let 5r^, (x) and Sr^,, (x) 

 be the corresponding shapes of the cavity for some 

 value of a. In each case the point of separation 

 is at x = 0. We choose A > so that the amplitude 

 of A6rcj '^) at X = " equals the amplitude of i5r (x) . 

 (Notice that their wavelength is already the same.) 

 Next we choose a shift, s > 0, so that 



lim Aor^,, (x+s) + 6r , (x) = 0. 



c2 ' 



(106) 



6r (5) = k(5) f (5), 

 c 



where : 

 k(5) = Kjdcl) [(a Ki(U|) - Ul Ko|c|)] 



359 

 (A,l) 



(A, 2) 



Here we have chosen the unit of length to be equal 

 to r^, so that r^ = 1. We have to solve (A,l) with 

 f (x) = for X > and with 6r^(x) being prescribed 

 for X < 0. 



In order to apply Wiener-Hopf-Technique, we need 

 a multiplicative decomposition of k(5). We define: 



H(C) 



-k(C) (C^ - E.^ ) {g^ + 1) 







-1/2 



(A, 3) 



where C is the root of (47) . This function is 

 continuous and positive in -«» < C < "». By virtue 

 of well-known asymptotic expressions for the 



Bessel functions, K and Ki , we have: 



■' 



H(C) 



1 + (1/5) . C -* + 



(A, 4) 



Hence, we can decompose H(C) in the usual way, see 

 for instance Noble (1958) ; we find: 



H(C) = H (C) / H (C) 



(A, 5) 



where : 



Now we construct a shape, 6r (x) , which induces no 

 waves at infinity, as follows: 



6r (x) = A6rj^j(x+s) + 6rj^2 1'^' ' x < -s 



(107) 



6r (x) = A6r^j (x+s) + Sr-^^{x) . 



-3 < X < 



The dimensionless load f was introduced in (22). 

 By virtue of the linearity of our equations we 

 obtain the load corresponding to the shape (107) by 

 shifting the load due to Srj^j over a distance s, 

 multiplying it by A and summing the load due to 



6r, 



h2- 



This load is nonnegative in x < and 



vanishes in x > 0. The condition for the point of 



separation described in the preceding section is 



satisfied at x = 0. The shape of the cavity is 



obtained by a similar construction as for the load 



f . The value of Sr (x) tends to zero for x ->• «> by 



virtue of (106) . "^ 



Using this method we have constructed five 



functions 6rjj(x) which induce no waves at x = " for 



five different values of a respectively. The 



functions 6rj^j and i5rj^2 ^^^ th^ functions plotted 



in Figures 4 and 5 respectively. The results are 



plotted in Figure 7. The point of separation is at 



X = 0. The value of 6r, (x) in x > is unessential. 



h 



H (5) = exp 



i- r - In 



IT J + 



C 



[H(£;): 



d£ 

 C-5 



(A, 6) 



This represents two equations; the upper or the 

 lower of the ± signs must be read. The contour of 

 C"*" (resp. C") is the real axis, indented into the 

 upper (resp. lower) half of the complex C-plane at 

 t = £. The function K^ (g) [resp. H~(5)] is analytic 

 in the upper (resp. lower) halfplane. Using (A, 2-3) 

 and (A, 5) we can write (A,l) as 



6r^(5) (^2 - g2) (5+i) 1/2 jj+jgj 



1/2 - 

 •(5-i) H (5) f (S) . 



(A, 7) 



The function (C+i)"^ has a cut from -i to -i" and 



(5-i)l/2 j^^3 3 ^u^. fj-om i to i". They are both 



chosen so that they are positive for 5 ^ ". 



The function Sr (x) is prescribed for x < 0. 

 c 

 First we assume: 



(x) = e 



Ax 



< 



(A, 8) 



for some positive A; later we discuss the general 

 case. Fourier transformation gives: 



APPENDIX 



6r (5) = -i (2TT) ^/2 (C-iA)"-^ + fir "^(C), (A, 9) 

 c c 



SOME RESULTS DERIVED BY USING WIENER-HOPF-TECHNIQUE 



where 



In equation (73) we let the path of integration 

 be from -<*> to <» and we assume the load f (x) to 

 vanish for x > 0. Then applying Fourier transform 

 (34) to both sides of this equation, we obtain: 



5r "^(C) = (2iT) ■'■''^ 

 c 



iCx 



e 6r (x) dx (A, 10) 

 c 



