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

1954 ◽  
Vol 50 (2) ◽  
pp. 319-333 ◽  
Author(s):  
P. C. Clemmow ◽  
J. Heading
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Point Of View ◽  
Successive Approximations ◽  
Horizontal Magnetic Field ◽  
First Order ◽  
Coupled Equations ◽  
Oblique Propagation ◽  
Special Cases ◽  
Magnetic Meridian

ABSTRACTIt 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.

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

1957 ◽  
Vol 53 (3) ◽  
pp. 669-682 ◽  
Author(s):  
K. G. Budden ◽  
P. C. Clemmow
Keyword(s):  
Magnetic Field ◽  
Electromagnetic Waves ◽  
Second Order ◽  
Successive Approximations ◽  
Horizontal Magnetic Field ◽  
Numerical Work ◽  
First Order ◽  
Coupled Equations ◽  
Special Cases ◽  
Chief Interest

ABSTRACTThe four first-order ‘coupled’ equations governing the propagation of electromagnetic waves in the ionosphere, previously obtained in symbolic matrix form (Clemmow and Heading (4)), are expressed explicitly in terms of the ionospheric parameters. The physical significance of the equations is illustrated by considering the energy flux in one characteristic wave when coupling and damping are neglected. Three special cases are then discussed for which second-order coupled equations are also given, namely, the cases of (a) vertical incidence with oblique magnetic field, (b) oblique incidence with vertical magnetic field, (c) horizontal magnetic field in the plane of incidence. For case (a) the second-order equations are those previously derived by Försterling(5).The form of the coupled equations is physically illuminating and, in principle, suitable for solution by successive approximations. Extensive numerical work has indeed been carried out on the second-order coupled equations in case (a) (e.g. Gibbons and Nertney(6)), and it is probable that the first-order coupled equations would prove more advantageous. The present authors, however, feel that better methods are available for purely numerical work (e.g. Budden(3)), and that the chief interest of the coupled form is that it shows the scope and limitations of the physical conception of characteristic waves.

scholarly journals Oblique propagation of electromagnetic waves in a slowly-varying non-isotropic medium

1936 ◽  
Vol 155 (885) ◽  
pp. 235-257 ◽  
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Electron Density ◽  
Isotropic Medium ◽  
Electromagnetic Waves ◽  
Mathematical Theory ◽  
Stratified Medium ◽  
Oblique Propagation ◽  
Isotropic Media ◽  
Mathematical Justification

The influence of the earth’s magnetic field on the propagation of wireless waves in the ionosphere has stimulated interest in the problem of the propagation of electromagnetic waves through a non-isotropic medium which is stratified in planes. Although the differential equations of such a medium have been elegantly deduced by Hartree,f it appears that no solution of them has yet been published for a medium which is both non-isotropic and non-homogeneous. Thus the work of Gans and Hartree dealt only with a stratified isotropic medium, while in the mathematical theory of crystal-optics the non-isotropic medium is always assumed to be homogeneous. In the same way Appleton’s magneto-ionic theory of propagation in an ionized medium under the influence of a magnetic field is confined to consideration of the “ characteristic ”waves which can be propagated through a homogeneous medium without change of form. In applying to stratified non-isotropic media these investigations concerning homogeneous non-isotropic media difficulty arises from the fact that the polarizations of the characteristic waves in general vary with the constitution of the medium, and it is not at all obvious that there exist waves which are propagated independently through the stratified medium and which are approximately characteristic at each stratum. The existence of such waves has usually been taken for granted, although for the ionosphere doubt has been cast upon this assumption by Appleton and Naismith, who suggest that we might “ expect the components ( i. e ., characteristic waves) to be continually splitting and resplitting”, even if the increase of electron density “ takes place slowly with increase of height”. It is clear that, until the existence of independently propagated approximately characteristic waves has been established, at any rate for a slowly-varying non-isotropic medium, no mathematical justification exists for applying Appleton's magnetoionic theory to the ionosphere. It is with the provision of this justification that we are primarily concerned in the present paper. This problem has been previously considered by Försterling and Lassen,f but we feel that their work does not carry conviction because they did not base their calculations on the differential equations for a non-homo-geneous medium, and were apparently unable to deal with the general case in which the characteristic polarizations vary with the constitution of the medium.

scholarly journals Low-degree mixed modes in red giant stars with moderate core magnetic fields

2019 ◽  
Author(s):  
Shyeh Tjing Loi ◽  
John C B Papaloizou
Keyword(s):  
Magnetic Field ◽  
Perturbation Theory ◽  
Magnetic Fields ◽  
Differential Equations ◽  
Order Perturbation ◽  
First Order ◽  
Low Degree ◽  
Mixed Modes ◽  
The Magnetic Field ◽  
First Order Perturbation

Abstract Observations of pressure-gravity mixed modes, combined with a theoretical framework for understanding mode formation, can yield a wealth of information about deep stellar interiors. In this paper, we seek to develop a formalism for treating the effects of deeply buried core magnetic fields on mixed modes in evolved stars, where the fields are moderate, i.e. not strong enough to disrupt wave propagation, but where they may be too strong for non-degenerate first-order perturbation theory to be applied. The magnetic field is incorporated in a way that avoids having to use this. Inclusion of the Lorentz force term is shown to yield a system of differential equations that allows for the magnetically-affected eigenfunctions to be computed from scratch, rather than following the approach of first-order perturbation theory. For sufficiently weak fields, coupling between different spherical harmonics can be neglected, allowing for reduction to a second-order system of ordinary differential equations akin to the usual oscillation equations that can be solved analogously. We derive expressions for (i) the mixed-mode quantisation condition in the presence of a field and (ii) the frequency shift associated with the magnetic field. In addition, for modes of low degree we uncover an extra offset term in the quantisation condition that is sensitive to properties of the evanescent zone. These expressions may be inverted to extract information about the stellar structure and magnetic field from observational data.

First Order Partial Functional Differential Equations with Unbounded Delay

2003 ◽  
Vol 10 (3) ◽  
pp. 509-530
Author(s):  
Z. Kamont ◽  
S. Kozieł
Keyword(s):  
Differential Equations ◽  
Integral Functional ◽  
Functional Differential Equations ◽  
Generalized Solutions ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
First Order ◽  
Unbounded Delay ◽  
The Cauchy Problem ◽  
Functional Differential

Abstract The phase space for nonlinear hyperbolic functional differential equations with unbounded delay is constructed. The set of axioms for generalized solutions of initial problems is presented. A theorem on the existence and continuous dependence upon initial data is given. The Cauchy problem is transformed into a system of integral functional equations. The existence of solutions of this system is proved by the method of successive approximations and by using theorems on integral inequalities. Examples of phase spaces are given.

Unified Theories for Massive Spin 1 Fields

1973 ◽  
Vol 51 (14) ◽  
pp. 1467-1470 ◽  
Author(s):  
A. Shamaly ◽  
A. Z. Capri
Keyword(s):  
Differential Equations ◽  
First Order ◽  
Special Cases ◽  
Massive Spin ◽  
Unified Theories ◽  
Stueckelberg Formalism ◽  
First Order Differential Equations

A pair of sets of first-order differential equations is given. They contain as special cases all the known spin 1 theories except the Stueckelberg formalism. It is also known that all of these theories are essentially causal.

scholarly journals Chaos in models of double convection

1992 ◽  
Vol 237 ◽  
pp. 209-229 ◽  
Author(s):  
A. M. Rucklidge
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fluid Layer ◽  
Period Doubling ◽  
Thermosolutal Convection ◽  
Two Dimensional ◽  
Lorenz Equations ◽  
Third Order ◽  
Horizontal Magnetic Field

In certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosolutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil'nikov bifurcation (Proctor & Weiss 1990). This establishes, for the first time, the existence of chaotic solutions of the equations governing two-dimensional magneto-convection. Moreover, in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modelled by a new third-order set of ordinary differential equations, which is shown here to have chaotic solutions that are created in a homoclinic explosion, in the same manner as the chaotic solutions of the Lorenz equations. Unlike the Lorenz equations, however, this model provides an accurate description of convection in the parameter regime where the chaotic solutions appear.

Periodic solutions of ordinary differential equations with one-sided growth restrictions

1978 ◽  
Vol 82 (1-2) ◽  
pp. 95-106 ◽  
Author(s):  
J. Mawhin ◽  
W. Walter
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
First Order ◽  
Asymptotic Nature ◽  
Special Cases ◽  
Existence Of Periodic Solutions ◽  
Growth Restrictions ◽  
Functional Differential ◽  
The Right

SynopsisThe existence of periodic solutions is proved for first order vector ordinary and functional differential equations when the right-hand side satisfies a one-sided growth restriction of Wintner type together with some conditions of asymptotic nature. Special cases in the line of Landesman-Lazer and of Winston are explicited.

scholarly journals Some of the methods used to solve complete and incomplete differential equations

2021 ◽  
Vol 12 (6) ◽  
pp. 193-200
Author(s):  
BASHEER ABD AL-RIDA SADIQ
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Second Order ◽  
First Order ◽  
Special Cases

This paper studies the methods used to solve complete and in complete differential equations and types of first order and second order and Exact differential equation to solve integration general in This equation Fur there more, and the Special cases to find the integration factor use solve those types of equations is use as well,supported by a relevant variety of examples.

Identities between reflexion and transmission coefficients and electric field components for certain anisotropic modes of oblique propagation

1971 ◽  
Vol 6 (2) ◽  
pp. 257-270 ◽  
Author(s):  
J. Heading
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Differential Equations ◽  
Electric Field ◽  
Electromagnetic Waves ◽  
Electric Polarization ◽  
Stratified Medium ◽  
Transmission Coefficients ◽  
Oblique Propagation ◽  
Integral Identities

A wide-ranging investigation is rendered possible by a judicious combination of products of electric field components and electric polarization components for two distinct modes of propagation of electromagnetic waves in an anisotropic, ionized, stratified medium. The differential equations, governing oblique propagation in these two distinct modes in such a medium, are combined to yield various integral identities when integrated throughout the medium. These lead to a large number of relations between the reflexion and transmission coefficients (for incidence from below and from above) and the fields throughout the medium, each containing as a factor just one of the components of the external magnetic field pervading the medium.

The method of finite differences for nonlinear functional differential equations of the first order

2020 ◽  
Vol 27 (4) ◽  
pp. 605-616
Author(s):  
Elżbieta Puźniakowska-Gałuch
Keyword(s):  
Weak Convergence ◽  
Differential Equations ◽  
Finite Differences ◽  
Initial Conditions ◽  
Functional Differential Equations ◽  
Successive Approximations ◽  
Duality Principle ◽  
Nonlinear Functional ◽  
First Order ◽  
Functional Differential

AbstractNonlinear functional partial differential equations with initial conditions are considered on the cone. The weak convergence of a sequence of successive approximations is proved. The proof is given by the duality principle.

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