254 



EXPONENTIAL M CURVE IN NONSTANDARD PROPAGATION 



WS — 



+ 03 



etc (62) to more general types of M anomalies. To begin with, 

 if 



where the value of Di/a obtained from equation (60) 

 is used in equations (61) and (62). Next, one substi- 

 tutes these values of the C's in the right-hand sides 

 of equations (56) to (59) and resolves for Di/a and 

 the C's. This procedure has been found to be rapidly 

 convergent and is, furthermore, self-correcting in 

 case of arithmetical errors. 



257 EXPANSION OF D t INTO A 



POWER SERIES IN a 



When a is small, it is convenient to expand D k 

 into the series 



D k = £>*«» + a D k ™ + a- ZV 2 > + • • • . (63) 

 It is known from standard perturbation theory that 



Pmk 



—i It 



(r m — 7-jO ' 

 to ± k . (64) 



It is possible also to derive an alternative expression 

 forZ)^: 



1 



D K ^ (X) =^fl t <»W J + ^=e 



z v x 



- D k ° x - (l 3 /l2) . 



1 D.OX- 



X3 



l + ^)D^(\ + x)+D^(\) 



D k (1) (x)\Vx dx 



i 



= xzv» (x)2 + __ e 



D»<« (X) + ( 2 + -) ZV» (X -j 



1 ^(0) x + (X 3/ 12) 



x) K/.r 



dx . (65) 



Since the former expression is simpler for compu- 

 tational purposes, we shall not give here the deriva- 

 tion of equation (65). 



258 APPLICABILITY OF PERTURBATION 

 METHOD TO A MORE GENERAL CLASS 

 OF M ANOMALIES 



It is possible to apply the results obtained for the 

 case when the M anomaly is of the form f(z) = ae -Xz 



f{z) 



+ ye~ 



(66) 



then we merely write in equation (28) in place of 

 «/WX), [aj3„ m (X) + Y&imG")]- Once the /?„ m (X) are 

 computed as functions of X, there is no additional 

 labor required to deal with an /(z) which consists of 

 a sum of any number of exponential terms. If instead 

 of f(z) = ae~ z we had f(z) = aze~ z , then the corre- 

 sponding /3', im (X) would be 



/3' nm (X) = U m °(z) U n \z)ze-^dz = 



dAm (X) 

 rfX 



(67) 



If (3„ m (X) is known, d/3„ m (X)/dX can be computed 

 directly from equation (43). When equation (43) is 

 integrated numerically, the derivative dfi nm (\)/d\ is 

 computed at each point in any case. Evidently, for 

 f(z) = az k e~^, where fc is a positive integer. 



(68) 



Pmm (X) = U m %z) U n \z) Z" e^ 2 dz 



= (-) 



„ d k p nm (X) 

 d\ k 



By successive differentiation of equation (43), it 

 is possible to express any high order derivative of 

 /3 nm (X) in terms of /3 rem (X). From a purely formal point 

 of view we can say therefore that by our method we 

 can treat any M anomaly by expanding it into a 

 series of Laguerre functions, since these functions 

 involve only terms of the form z k e~' Kz . It may be 

 pointed out that a single term z k e~ Xz vanishes both 

 at the ground and at great height and reaches a 

 maximum at z = fc/X. Such a single term is therefore 

 suitable to represent an elevated duct. 



259 COMPUTATIONAL PROGRAM 



FOR THE EXPONENTIAL MODEL 



The Analysis Section of the Columbia University 

 Wave Propagation Group has undertaken the com- 

 putation of /3 nm (X) for X = 0(0.1)4.0 and n, m = 

 1, 2, 3, 4, 5. With these functions tabulated, it is 

 planned to compute the characteristic values D k for 

 such values of a and X that the difference between the 

 values of D k obtained from the fourth order deter- 

 minant and from the fifth order determinant will be 

 only about 0.01. The program also calls for the 

 computation of the height-gain functions from equa- 



