RECTANGULAR W AVE Gi IDES \55 



were evaluated by Simpson's rule. The second derivative of A'"""* was 

 computed from the even order central (Hflferences of A' . For a = 1/6, 

 corresponding^ to an anj^le of tt 3 Ijetween tlie two sides of the horn, calcula- 

 tions at two representative wave lengths led to the table 



(9-4) 



An idea of the variation of A may be ol^tained from its values at — ^, 

 — .6, 0, .6, 1.8, 3.6 which are approximately .67, .76, .98, 1.62, 4.56, 17.4, 

 respectively. The range of integration was — 3 < r < 4.4. 



The second method of computation used the formulas (3-11) with Fi 

 playing the role of y. The differential equation (9-1) was integrated by the 

 Kutte-Runge method, the interval between successive values of z- being 

 0.2. For c = .8173 the values obtained were 



vx V2 y T Rh (9-5) 



-.6 .6 -.202 - /.981 -.142 - /.794 -.0167 + /.0658 



-1.2 1.2 -.218 - /1.004 -.049 - /.696 -.0525 + /.0754 



-1.8 1.8 -.225 - /.989 +.086 - /.716 -.0512 + /.0753 



-2.4 2.4 -.220 - /1. 000 .136 - /.842 -.0424 + /.0722 



In order to gain an idea of the meaning of these values of v it should be 

 recalled that ic — v + id and the walls of the guide are at ^ = and 9=7:. 

 An interval of length tt = 3.14 ■• • in the v direction therefore corresponds 

 roughly to a distance equal to the width of the guide. The above table 

 indicates that, loosely speaking, most of the reflection occurs close to the 

 junction of the horn and wave guide. 



The last value of Rh in (9-5) agrees quite well with the value —.0420 + 

 /.0724 obtained from the approximate expression (9-3). It appears that the 

 method leading to (9-5) is superior to the one based on (9-3) since, in theory, 

 it may be made as accurate (insofar as the single equation (9-1) may replace 

 the set of equations (2-4, 5)) as desired. Moreover, less actual work seems 

 to be required. 



The approximation (7-1) yields, for c = .8173, 



^"~ 2? ~ 2X8173? " '•^''^ 



which is considerably in error, as we might expect, since a = 1 6 is not small. 

 However, if we use the a{)i)roximation (7-13) and evaluate the integral 

 by Simpson's rule we obtain 



