Chapter 25 



PERTURBATION THEORY FOR AN EXPONENTIAL M CURVE 

 IN NONSTANDARD PROPAGATION 1 



28. 1 



ABSTRACT 



In this chapter a perturbation method is developed 

 for treating nonstandard propagation in the case 

 when the deviation of the M curve from the standard 

 (= the M anomaly) can be represented by a term 

 ae~ u , where z denotes height in natural units. The 

 method is also applicable to other forms of the M 

 anomaly which can be derived from an exponential 

 term by differentiation with respect to X; in fact, in 

 its region of convergence, it is formally applicable 

 to the most general type of M curve, including 

 elevated ducts. The region of practical convergence 

 of the method ranges from standard down to cases 

 where the decrement is a small fraction of the 

 standard value. 



The procedure followed is to express the height- 

 gain function U k (z) of the fc-th mode in the non- 

 standard case as a linear combination of the height- 

 gain functions U m °(z) of all the modes in the standard 

 case. 



U k {z) 



-S 



A km U m \Z) . 



(1) 



The execution of this plan hinges on the possibility 

 of evaluating the quantities 



,(X) 



= i U " 



(z) UJ(z) e-»dz . (2) 



It is shown that p„ m (\) satisfies the differential 

 equation 



d\ 2X 



.(X) 



\_~k + l {Dm0+Dn ° )+ l +ib (2v ~ zv) *] ' 



(3) 



whose solution is 



X3 1 



1 - (23 ° + 23 0) + ±--l (D -23 0) 



z Vx 



/ dx -?(D + 23 0)-E + L(D 0-D °)2 



Jo -\/x 



(4) 



Here D m ° denotes the characteristic value of the 

 m-th mode in the standard case. For large X the 

 following asymptotic formula holds 



fx 3 + 2X (D ra ° + TV) - 2 + i (D; - D.»)»l 

 8 T3X 3 + 2X (DJ + ZV) - 1 (DJ - ZV) 2 ] 



(5) 



X 3 + 2X (D„ 



■ D n °) -2 +-(1V -D n °) 2 



Having determined the fi nm (\) from equation (4), 

 or by a numerical solution of equation (3), the 

 characteristic values D k and the coefficients A km are 

 to be solved from the infinite system of equations 





A km (D k - DJ) 8 nm + a/3 nm (X) 



]-.. 



n = 1,2,3 • • • (6) b 



For this purpose a simple iterative procedure has 

 been developed, which has been found to be rapidly 

 convergent. The A km are normalized by the condition 



I 



i ,?{z)dz= 1 =J^A km - 



m-l 



(7) c 



One can also expand D t as a power series in a 

 D k = D° k + aDl + a"-Dl + • • • , 



£* (1) = -/3™;Z>*< 2 >=e™ 



So 



Pmk 



r k ) 



m ±k . 



An alternative expression for D c f is given in equa- 

 tion (65). 



a By C. L. Pekeris, Columbia University Wave Propagation 



Group. 



b 5„ m = 1, n = m 



!>nm = 0, n ± m . 

 r co 

 The integral I Uk 2 (z)dz diverges when taken along the 



real axis; it converges, however, and to the same limit, when 

 the path is a radial line in the fourth quadrant of the z plane. 

 In the sequel, whenever an integral is divergent it will be 

 understood that the path is suitably modified. 



249 



