Perturbation Theory of the Normal Modes for an Exponential M-Curve in Non-Standard Propagation of Microwaves

Abstract

In this paper a perturbation method is developed for treating non‐standard propagation of microwaves beyond the horizon in the case when the deviation of the M‐curve from the standard (≡ the M‐anomaly) can be represented by a term αe^−λz, where z denotes height in natural units. Here M denotes the modified index of refraction of the air. 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 λ; 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 highly substandard conditions down to cases where the decrement is a fraction of the standard value. The procedure followed is to express the height‐gain function U_k(z) of the kth mode in the non‐standard case as a linear combination of the height‐gain functions U_m⁰(z) of all the modes in the standard case: $$U_k\mathrm{(z)=}{\displaystyle \sum _{\text{m=1}}^{\infty }}{A}_{\mathrm{km}}{U}_m^0\mathrm{(z)}$$ . The execution of this plan hinges on the possibility of evaluating the quantities $${\beta }_{\mathrm{nm}}\mathrm{(\lambda )=}{\displaystyle {\int }_0^{\infty }}{U}_n^0{\mathrm{(z)U}}_m^0{\mathrm{(z)e}}^{\mathrm{-\lambda z}}\mathrm{dz}$$ . It is shown that β_nm(λ) satisfies the differential equation $$\frac{{\mathrm{d\beta }}_{\mathrm{nm}}}{\mathrm{d\lambda }}=\frac{2}{\text{2λ}}{\mathrm{+\beta }}_{\mathrm{nm}}\mathrm{(\lambda ).}\left[-\frac{1}{\text{2λ}}+\frac{1}{2}{\mathrm{(D}}_m^0{\mathrm{+D}}_n^0\mathrm{)+}\frac{{\lambda }^2}{4}+\frac{1}{{\text{4λ}}^2}{\mathrm{(D}}_m^0{\mathrm{-D}}_n^0{)}^2\right]$$ , whose solution is $$\begin{array}{c}{\beta }_{\mathrm{nm}}\mathrm{(\lambda )=}\frac{1}{\text{2√λ}}\exp \left[\frac{\lambda }{2}{\mathrm{(D}}_n^0{\mathrm{+D}}_m^0\mathrm{)+}\frac{{\lambda }^3}{12}-\frac{1}{\text{4λ}}{\mathrm{(D}}_m^0{\mathrm{-D}}_n^0{)}^2\right]\\ ·{\displaystyle {\int }_0^{\lambda }}\frac{\mathrm{dx}}{\mathrm{\surd x}}\exp \left[-\frac{x}{2}{\mathrm{(D}}_m^0{\mathrm{+D}}_n^0\mathrm{)-}\frac{x^3}{12}+\frac{1}{\text{4x}}{\mathrm{(D}}_n^0{\mathrm{-D}}_m^0{)}^2\right]·\end{array}$$ Here D_m⁰ denotes the characteristic value of the mth mode in the standard case. For large λ the following asymptotic formula holds

Keywords

• SATISFIABILITY
• POWER SERIES
• PERTURBATION THEORY
• NORMAL MODES
• SYSTEM OF EQUATIONS
• INDEX OF REFRACTION
• DIFFERENTIAL EQUATION

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