Coupled forms of the differential equations governing radio propagation in the ionosphere

Abstract

It is shown that the equations governing oblique propagation in a horizontally stratified ionosphere with an oblique magnetic field can be cast into a form suitable for solution by successive approximations. In the general case this ‘coupled’ form consists of four first order differential equations, each being associated with one characteristic wave. In the special cases (a) horizontal magnetic field, (b) plane of incidence perpendicular to the magnetic meridian, two second order coupled equations of a particular type can be derived, each of which is associated with a pair of corresponding upgoing and downgoing characteristic waves. These latter equations are similar to, and include, those already given by Försterling(7) for vertical incidence.The cases in which there is no coupling are briefly considered from the point of view of the first order equations.The general formulation provides a basis for assessing the validity of the standard ‘ray’ approximation, alternative to that developed by Booker (2), and brings out the nature of its breakdown in coupling and reflexion regions.Specific applications and extensions of the theory are left for later consideration.

Mode Coupling in the Solar Corona. IV. Magnetohydrodynamic Waves

“…From a formal viewpoint we have seen that it is straightforward to treat the general case of coupling between MHD waves obliquely incident on a stratified medium using the method of Clemmow and Heading (1954). We have used this method to show that fast Alfven-mode coupling for VA ~ C s is equivalent to magnetoacoustic Alfven-mode coupling in cold plasma theory (Part III), as one would expect.…”

“…From a practical viewpoint, application of Clemmow and Heading's (1954) method to obliquely incident MHD waves leads to an excessively cumbersome theory. The treatment of coupling between all three modes near the double coupling point at sin 8 = 0 and C s = V A has not been pursued here simply because of the lengthy algebra involved.…”

“…It is convenient to generalize the theory developed by Clemmow and Heading (1954) and in Parts I and II of this series (Melrose 1974a(Melrose , 1974b as follows.…”

Perturbation Methods Applied to the Reflection of Radio Waves From the Ionosphere

“…2) First order coupled equations 4 Clemmow and Heading present a set of first order coupled equations for the quantities ExS E, H x and H The equations…”

Magneto-ionic propagation in homogeneous media. Part I. Transverse propagation

“…In other words , linear co mbinati ons of the type [Ex +ip(z)E y] or [a tEx + bIHx+a2E y+ b2Hy] obeyin g se parate (decoupled) wave equations do not exis t for inhomogeneous magne to-ionic m edia. F orste rling [1942] , Rydb eck [1950], and lately Clemmow and Heading [1954] , tried to determine s uitable lin ear co mbination s of the horizo ntal co mpone nts of th e fi eld vectors for whi ch the couplin g func tion would re main small throughout th e layer to justify th e application of pe rturbation me thods. It can be s hown that the co uplin g functions of th ese equations sati s fi ed by th e specific lin ear combinations s uggested by th ese workers have precisely the same numb e r and types of poles as L(z) in (2) has.…”