Hr* i/ k(1 - *"> " * f ,„, 



y = ; (12a) 



1 _ gtx? " 



zf k(1 - (T a ) : 



In the majority of practical cases, the value of the ratio #/n lies between 

 2/3 and 1/3. Placing */n = 1/2 in Eq. (11c) and substituting (11c) in (12a) we ob- 

 tain the simple expression 



y = 1 ' 718 ~ ^L (a) 



1 - 0.374*yx 



With ocjn = 2/3 we obtain the slightly altered equation 



„ .. 1.724 <c ffic 1 " ( . v 



1 - 0.365 ocVx 



while <x/n = 1/3 gives 



. 1-715 ~ VF (0) 



1 - 0.382«^yx 



Any number of similar equations may be obtained by using different values of 

 the ratio <*/n. As a limiting case o(/n = gives the equation 



. 1.714 ■* y? (4) 



1 - 0.389 ocyx 



which differs but slightly from (c). 



If equation (d) be expanded in a series, the first two terms are identical 

 with the first two terms of equation (12). Moreover, the expansion of any of the 

 equations (a) to (d) yields a series all terms of which are positive. Therefore, 

 the same conclusion can be drawn from these equations as from Eq. (12) concerning 

 the impossibility of nodal points between points of support in the longitudinal 

 plane. It is evident that Eq. (a) - (c) are much better approximations to the 

 envelope than Eq. (12). Each gives extremely accurate values in that particular 

 region of et/n for which it was developed and good values for a considerable range 

 of ct/n. 



To determine just how closely these approximate equations check with Eq. 

 (6), about 70 values of y, covering the region x = 4/3. 10~ 6 to 12.10" fo (t/d = 0.002 

 to 0.006) and OC = 1/4 to 16 (1/d = 6.28 to 0.1), were computed by each of the six 

 equations (6), (7), (a), (b), (c), and (d). These values and their percentage 

 variation from the exact values computed by Eq. (6) are given in Tables II and III, 

 page 19. 



