Asymptotic Solutions for the Normal Modes in the Theory of Microwave Propagation
Authors: Pekeris C. L. 
Published: January 2005

In this paper are presented several extensions of the ``W.K.B.'' method in the asymptotic solution of differential equations. The developments were made in the course of a systematic study of the application of asymptotic solutions to the normal-mode theory of microwave propagation. In Section 1 a brief outline is given of the now standard normal-mode theory of propagation of microwaves in an atmosphere with a horizontally stratified index of refraction. Particular emphasis is given in this study to the case of a ``surface duct.'' The case of ``leaky modes'' is studied in Section 2. Here the aim has been to obtain explicitly the terms after the leading one in the asymptotic expansion of the solution, in order to have an estimate of the order of magnitude of the error introduced by the use of the leading term only. This is achieved in Eq. (20), which is of general applicability to a wide class of physical problems. The first correction term to the phase-integral solution for the characteristic values of the normal modes is set out in Eqs. (28) to (30). An alternative asymptotic solution for the case of leaky modes, which includes the first correction terms, is given in Eqs. (30), (40), and (41). In Section 3 a direct method is given of determining the characteristic values from the phase-integral relation. If the modified index of refraction y(h) is given by a power series (49) then with the aid of Eqs. (50)–(52) one can compute directly the characteristic value Λm by Eq. (46), using (47) and (48). The degree of success of this method is illustrated in Table I. The principal contribution of this study is in the development of asymptotic solutions for the case of transitional modes which are on the border line between the leaky modes and the trapped modes. The standard ``W.K.B.'' method is inapplicable in this case. The results for the determination of the characteristic values and height-gain functions are as follows. One first obtains ν0 defined in (85), which in the case of an exponential model can be facilitated by the use of Fig. 2. The quantities γm0, dγm0/dν0, and [(∂P(ν0)/∂ν) . (∂P(ν0) / ∂γm0)] are then read off Table IV. The characteristic value is then determined from (87), (88), and (68), particular examples of which are given in (89) and (90) for the ``exponential model,'' and in (94) and (96) for the ``power law'' model. These include a correction term Δm which was obtained from the perturbation theory of transitional modes developed in Section 4D. A comparison between exact characteristic values and those obtained from the theory of transitional modes is shown in Tables II and III for the ``exponential'' and ``power law'' models, respectively. It is seen that the theory of transitional modes developed here yields accurate characteristic values over a wide region, extending into the leaky modes on one side and into the trapped modes on the other side. The height-gain functions can be obtained from (60), (69), (76), (77), (83), and (101). In the case of an exponential model the variables u and v can be computed with the aid of Fig. 2. The asymptotic theory of trapped modes developed in Section 5 follows Langer's method, whereby the difficulty with the Stokes phenomenon is completely obviated. The procedure used is to develop the solutions around the two turning points h1 and h2 shown in Fig. 1 and to join the solutions at the duct height h0. The relation determining the characteristic values thus obtained is (140), which bears a similarity to the corresponding equation for a ``bilinear'' model. Indeed, Eqs. (131), (132), (136), and (137) give the transformation required to turn the model with a continuous y(h) curve shown in Fig. 1 into one where y(h) is represented by two straight lines that intersect at h0. It is shown in Eq. (159) how one can derive the Furry-Gamow formula for the decrement of completely trapped modes from Eq. (140). Equations (160) to (163) give expressions for the height-gain functions for highly trapped modes. These cover the whole range of height including the vicinity of the two turning points.