A Discussion on the early days of ionospheric research and the theory of electric and magnetic waves in the ionosphere and magnetosphere - Phase integral methods for studying the effect of the ionosphere on radio propagation

The phase integral method is a form of ray theory, extended to use complex values of the space coordinates. Its application to radio propagation studies was pioneered by T. L. Eckersley who showed how to use it for calculating (a) the reflexion coefficient of the ionosphere, (b) the propagation constant for radio waves guided by the Earth’s surface and by the ionosphere or troposphere, and (c) the coefficient for coupling of an ordinary and an extraordinary wave in the ionosphere. The method involves the evaluation of integrals along suitably chosen contours in complex space. It is approximate but often capable of high accuracy and often quicker to use than more exact methods. Its justification is based on the physical principles of analytic continuation and of uniform approximation. For reflexion and coupling problems in a horizontally stratified ionosphere, the contours used for the phase integrals are determined by those real or complex heights called ‘reflexion’ or ‘coupling’ points, where two roots of the Booker quartic equation are equal. The study of the behaviour of the governing equations near these points shows when failure of the phase integral method may be expected.

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Theory of the reflexion of low-frequency radio waves obliquely incident on the ionosphere

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1974.0034 ◽

1974 ◽

Vol 336 (1607) ◽

pp. 525-542 ◽

Cited By ~ 4

Keyword(s):

Low Frequency ◽

Extraordinary Wave ◽

Meridian Plane ◽

Radio Waves ◽

Quartic Equation ◽

Obliquely Incident ◽

Full Wave ◽

Magnetic Meridian ◽

Electron Gyrofrequency ◽

Wave Normal

The properties of the two principal reflexions for radio waves obliquely incident on a horizontally stratified ionosphere in and near the magnetic meridian plane, for frequencies less than the electron gyrofrequency, are investigated using 'full wave’ numerical methods. When the wave normal of the incident wave is close to either of two directions, which are in the magnetic meridian plane, at particular angles θ b and θ e to the vertical, then for propagation from south to north (northern hemisphere), the polarizations of the two reflexions are found to take anomalous values. This behaviour is related to the properties of the Booker quartic equation. An extraordinary wave incident at an angle near θ e in the N–S direction generates some of the upgoing ‘whistler’ mode, an d this process is also investigated.

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Theory of the modulation imposed on an obliquely incident radio wave reflected in a disturbed region of the lower ionosphere

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1976.0089 ◽

1976 ◽

Vol 349 (1659) ◽

pp. 555-570 ◽

Cited By ~ 3

Keyword(s):

Integral Method ◽

Collision Frequency ◽

Lower Ionosphere ◽

Electron Collision ◽

Disturbed Region ◽

Quartic Equation ◽

Obliquely Incident ◽

Electron Collision Frequency ◽

Reflexion Coefficient ◽

Phase Integral

A powerful disturbing wave enters the lower ionosphere and causes a periodic modulation of the electron collision frequency. A simple model is adopted for this disturbed region. The modulation transferred to an obliquely incident wanted wave that is reflected in or near it is investigated. The reflexion coefficient of the wanted wave is found by applying the phase integral method. The complex reflexion height of the wanted wave is a function of time in the modulation cycle. Results are discussed first for an isotropic ionosphere and are then extended to include the effect of the Earth’s magnetic field, and the calculation uses the Booker quartic equation. It is shown that the phase integral method is admirably suited to solve this kind of problem. Some examples are given to illustrate that the greatest amount of modulation is transferred when the wanted wave is reflected near the most disturbed part of the ionosphere. The relation of this to some observed effects near sunrise is discussed.

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Coupling points of the Booker quartic equation for radio wave propagation in the ionosphere

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1974.0016 ◽

1974 ◽

Vol 336 (1605) ◽

pp. 229-250 ◽

Cited By ~ 6

Keyword(s):

Wave Propagation ◽

Radio Wave ◽

Radio Wave Propagation ◽

Radio Propagation ◽

Electron Collisions ◽

Quartic Equation ◽

Integral Methods ◽

Northern Latitudes ◽

Electron Gyrofrequency ◽

Magnetic North

For a horizontally stratified ionosphere the four roots of the Booker quartic equation can often be used to give four independent W. K. B. type solutions of the electromagnetic equations which govern radio wave propagation, but this is only possible if the four roots are distinct. There are points in the complex height plane, called ‘coupling points’, where two roots of the quartic are equal, and in phase integral methods it is necessary to know their positions, because near them the W. K. B. solutions fail. There are eight coupling points which are important in radio propagation and four of these are of particular interest. Their positions are found from an equation given by Pitteway (1959). Their behaviour, including their loci as the azimuth and elevation of the incident wave are varied, is studied. It is illustrated by a specific example, for temperate northern latitudes, in which the frequency is one-quarter of the electron gyrofrequency and the azimuth of propagation is (magnetic) North to South or near to this. The effect of neglect or inclusion of electron collisions is investigated.

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The coalescence of coupling points in the theory of radio waves in the ionosphere

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1974.0169 ◽

1974 ◽

Vol 341 (1624) ◽

pp. 1-30 ◽

Cited By ~ 14

Keyword(s):

Electromagnetic Field ◽

Radio Wave ◽

Electromagnetic Fields ◽

Integral Method ◽

Radio Waves ◽

Coupling Coefficients ◽

Ionospheric Layer ◽

Partial Penetration ◽

Phase Integral ◽

Coupling Point

When a plane radio wave is obliquely or vertically incident on a horizontally stratified ionosphere, there are certain points in the complex height plane, called coupling points (which include reflexion points) where there is a breakdown in the independent propagation of the four waves, ordinary upgoing and downgoing, and extraordinary upgoing and downgoing. If a coupling point is far enough away from other coupling points and from singularities, it is said to be isolated and the electromagnetic field near it can be expressed in terms of Airy Integral functions. Then the phase integral method can be used with suitably chosen contours, to calculate reflexion coefficients and coupling coefficients. If two coupling points are too close together, however, the procedure needs modification. This paper studies the theory when two coupling points approach coalescence. It is confined to the cases where the same two waves are coupled at the adjacent coupling points, since this is the most important in practice. It is found that there are two kinds of coalescence called coalescences of the first kind C1, and of the second kind C2. For C1 the coupling remains strong when the coupling points move to coalescence. For C2 the coupling gets weaker, and disappears completely at exact coalescence. The electromagnetic fields near a point of coalescence can be expressed in terms of solutions of Weber’s equation, but the form of this equation is different in the two cases. The type C1 is important in the theory of partial penetration and reflexion for frequencies near the penetration frequency of an ionospheric layer. The type C2 is important in the theory of ʻcrossover’ for ion-cyclotron whistlers, and in the theory of the Ellis window and related phenomena. These applications are worked out as illustrations.

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