250 



EXPONENTIAL M CURVE IN NONSTANDARD PROPAGATION 



25.2 



INTRODUCTION 



In the theoretical treatment of nonstandard 

 propagation by the method of normal modes, one is 

 confronted with the task of solving the equation 



d*U„ 

 dh 2 



+ fc 2 



y(h) 



u m = 0, 



(8) 



the frequency band used, as will also the height 

 represented by one unit of z. 



263 FORMAL SOLUTION OF THE PROBLEM 

 BY THE PERTURBATION METHOD 



In order to solve the equation 



subject to the condition that U m (0) = and that 

 at h — > °° , U m should represent an upgoing wave 

 only. Here h denotes height in feet. 



y(h) = N 2 (h) - 1 = 2 X lO" 6 M(h) , k = — , (9) 



A 



and A m is the characteristic value which is generally 

 complex. It is convenient to introduce natural units 

 of height 



z = A , H = {k 2 q)~\ q = ^ = 2.36 X 10" 9 cm" 1 , 



\z + ae~^ + dA 



^—^-i j ■ ; ." ' I !> \ ( ,.::> •>, U.m 



we seek a solution in the form 



U k = J] A km UJ{z) , 



(16) 



H 



dh 



where U m °(z) are the height-gain functions of the 

 m-th mode in the standard case, which satisfy the 

 equation 



''"' " ' ' ' + DJ\ UJ(z) = 0. (17) 



(10) 



whereby equation (8) is transformed into 

 d 2 U., 



J n U m \z) I 2 dz = 



(18) 



dz 2 



+ 



z + }(z) + D„ 



U m (z) = . (11) 



The term/(z) in equation (11) represents the refrac- 

 tion anomaly and is equal to zero for a standard 

 atmosphere. In the first instance we shall be treating 

 the case where 



/(*) = ae~^ , (12) d 



and we shall later generalize the treatment to deal 

 with any M curve represented as a series of Laguerre 

 functions. If the original M curve is represented by 

 the expression 



MQi) = bh + aer ch , b = 0.036 ft" 1 , (13) 



then a and X are obtained as follows: 



The solutions of equations (17) and (18) are well 

 known : 



UJ(z) = C m ui H^ („) , u = | (z + Djy> ,(19) 



c m = W 



V3 



f 2 ) v v J l ( y '») _ J-i ("•») \ 



i - J-t «l"ll (20) 



_ _ -,12*73 



*-* ,« ■ - T m C 



m ' mp j ' m 



3w m \ § 



where 



Ji (Vm) + J-\ (.Vm) = . 



(21) 



(22) 



For small z the power series development of i7 m °(z) 

 is useful: 



'_» . ; Hi (-V a , X = cH 



(14) 



It is to be noted that in contrast to the constants a 

 and c in equation (13), which are independent of 

 frequency, the constants a and X in equation (12) 

 are frequency dependent. For a given observed M 

 curve the constants a and X will therefore differ with 



A t = - 



U m °(z) = iJ^A k z", 

 1 



k(k - 1) 



U m \z) = i 



DJA.-2 + A,_ 3 

 D° m 



(23) 

 , (24) 



; { -Q~) Z 12 



d No confusion should arise from the use of X in equation 

 (12) and the standard usage of X to denote wavelength. 



+.(£)■" + (&>•+■•■ 



(25) 



