355 



1/2 



1/2 



fl(x| 



0, 



C94a) 



fl(x) = (x - X2) / (x - X2) . X < X < xj . (94b) 

 1 2 



Finally f (x) vanishes for x , < x < and: 

 n n-1 - 



f{x) = (x-x )/(x -X ),x <x<x '^^^' 

 n n-1 n n-1 n - - n-1 



f (x) = 1 

 n 



X < X 



n 



(95b) 



These functions are plotted in Figure 3 . We approx- 

 imate the function f(x') in (89) by a linear com- 

 bination of the coordinate functions : 

 n 



f(x') = ICf (x') , (96) 



m=l m m 



where the C are unknown coefficients. In order to 

 approximate f(x') well near the origin, we have 

 chosen fi in a special way and, besides, the points 

 x are more densily distributed near the origin. 

 Since f{x') is almost constant for large negative 

 values of x' , we have chosen f to be constant in 

 (- ", X ) . 



Next we have to determine the coefficients Cj , 

 ..., C . We substitute (96) into (89) and then 

 the C must be chosen so that the difference 

 between the right hand side and the left hand side 

 of (89) is as small as possible, in the sense of 

 some norm. The computed values of the C appeared 

 to depend strongly on which norm was chosen for this 

 difference; many of these noinns give unreliable 

 results. We obtained reliable values of the C as 

 follows : 



Equation (89) with x = x , £=0,1,..., n-2 yields 



n 

 Z 



m=l 



M C 

 Im m 



= 6r (x + s), 5,= 0, ..., n-2 



where : 



M„ = f k(x„ - x') f (x' ) dx' . 

 Jim J I m 



(97) 



(98) 



At the points x.^-]^ and Xj^ we minimize the difference 

 between the right hand side and the left hand side 

 of (89) . The expression 



n 



r 



J,=n- 



m=l 



S-m 



- 61 



+ s)]' 



(99) 



We have checked these numerically computed values 

 of the Cjjj as follows. First, the computed approxi- 

 mation has the square root character (90) even in 

 the interval (X3,0). Second, the value of C^ equals 

 the right hand side of (91) within an error of 0.5%. 

 Third, if we replace the kernel k (x) in (89) by a 

 kernel k(x), which has the same behavior (69) at 

 the origin, and which for x ->■ " is also given by a 

 term A sin bx as in (56) , then (89) can be solved 

 effectively by the Wiener-Hopf method. If we apply 

 our numerical method to (89) with the kernel k, 

 then the numerically computed function, f , equals 

 the analytically computed solution within an error 

 of 1%. 



We have tried to compute the Cj, ..., C in 

 different ways; for instance: 



i) By collocating the points x , 



Xj^ with 



the exception of one point Xj^ , so that the number 

 of equations equals the number of unknowns. This 

 method had to be rejected because the computed 

 approximation for f (x) appeared to have oscillations 

 near Xj^ . 



ii) By minimizing the sum of squares of the 

 differences between the right hand and the left 

 hand side of (88) at the points Xq, ..., Xj^ . We 

 have also rejected this method, because oscilla- 

 tion occurred in f (x) near the origin. 



We make some remarks concerning the computation 

 of the matrix elements. M in (98). For m = 1, 

 . . . , n - 1 the integrand is non-zero only in a 

 bounded region. The kernel k (x) is written as the 

 sum of a logarithm and a function which is bounded 

 at x - 0. The integral over the logarithm is 

 evaluated analytically; the remaining term is 

 integrated numerically. For m = n the integrand is 

 non-zero in an unbounded region . For x £ Xj^ we 

 have fj^fx) = 1 and we must evaluate 



k(xi - x' ) dx' 



(100) 



Note, that the integrand does not tend to zero for 

 x' -> -", as follows from (65). However, the express- 

 ion (100) represents the deformation of the cavity 

 due to a loading which equals a step- function. This 

 deformation has been computed in (76,77). 



We now come to the determination of the shift, s. 

 The pressure in x < at r = r^ must exceed the 

 vapor pressure. Hence, by (22) with a = 0, we 

 must have f S 0, and by (90) , B > 0. As will be 

 shown in the Appendix, the shape of the cavity for 

 small values of x is given 



6r (x) 



6r^(0) + 6r (0)x - 4B {x'^/-n)^/3 , x 4- 

 h h 



is a quadrative , non-negative function of the C 

 Now the Cj, ..., Cj^ are determined so that they 

 minimize (99) with the constraints (97) . 



(101) 



This implies that the radius of curvature tends to 

 zero for x -I- . Since the fluid may not penetrate 



FIGURE 3. The coordinate functions 

 fi, . . . f . 



