Propagation of electromagnetic waves through a continuously varying stratified anisotropic medium

Abstract

Express ions arc developed for the transmiss ion and reflection coeffici ents for p ropa gation of a plane wave through a layered medium, taking account of the effects of t he static m agnetic fi eld . A matri x formulation is used which a llows proceed in g to t he limit of a continuo usly varyin g medium, a nd series expansions of th e field", for this case arc developed. The res ults a rc ex pec ted to have appli cation to interpretation of VLF d a ta obtained within a nd above the lower ionosph ere. IntrcductionTheoretical in.vestigations of VLF propagaLion have primarily b een concerned with the Tegion b elow the ionosphere, as evidenced by the use of semi-infinite model ionospheres not only in earli er semiquantitative studies [Budden , 1951] but also as a reasonable simplifyin g assumption in CUlTent studies [Wait and Walters, 1963fl, 1963b] . Even in studies in which a semi-infini te ionosphere is not ass umed, the transmission coefficien ts are often not developed [Wait, 1962a[Wait, , 1963 or flre developed as an incidental part of the analysis [Johler and Harper, 1962]. This emphfl,sis has been consistent with the nature of the available experimental data. However, as data from the ionosphere b ecome available [Rorden et al. , 1962], their in terpretation requires consideration of propflgation in to and through the ionosphere. It is t he purpose of this paper to provide a framework Jor this type of calculation.As is generally the case if tractability is to be m aintained, some simplification of the physicfll situation is desirable. A suitable model for our purposes is a stratified plane ionosphere with plane waves obliquely in cident. The earth's magn etic field is included since it is expected to allow propagation through the ionosphere (whistler mod es) at some fr equencies and angles of incidence. It is assumed to have constant strength but to be at an arbitrary angle to the plane of stratification.The equations are first developed for an arbitrary number of homogeneous layers, a problem considered by Johler and Harper [1962]. A different system of representation is used in th is study, allowin g the number of layers to be increased by multiplying together more and more matrices of constant (4 X 4) size rather than by increasing the size of a single matrix. This form has the advantage of allowing use of available computer subroutines to handle the matrices and the use of slow access storage for matrices not currently being operated upon. The valu es of the matrix elements are determined by the solution of the Booker quartic for each l ayer. The terminology used is that employed by Yabroff [1957] in treating a single planar boundary.In the next step, it is assumed that the layers become infinitely thin while becoming infinite in number , to represent a continuously varyin g medium. A series expansion for the tran smi ssion and reflection coefficients is then obtained. The reflection coefficienLs series have the form found by W a it [1962a] and by Heading [1963] by an iterative procedure. A phys...

An improved ray theory and transfer matrix method‐based model for lightning electromagnetic pulses propagating in Earth‐ionosphere waveguide and its applications

“…The field propagation effect in each layer is converted into a transfer matrix which involves the phase integration for each mode within the layer. The whole field at any height is then acquired by a series of matrix multiplication [Johler and Harper, 1962;Price, 1964;Inoue and Horowitz, 1966;Altman and Cory, 1969;Nagano et al, 1975;Nagano et al, 2003]. For the narrow-and high-frequency waves propagating in the ionosphere, the Wentzel-Kramers-Brillouin (WKB) approach is widely used [Budden, 1988].…”

“…In the model, the lower ionosphere is assumed to be a magnetized, anisotropic, collisional, and cold vertical stratified plasma. The transfer matrix method [ Price , ; Nagano et al ., ; Nagano et al ., ; Zhao et al ., ] is applied to calculate the propagation property of each ray inside the dispersive ionosphere and therefore the reflection coefficient of each ray. Only the first‐ and second‐order rays named first and second within the range of 1000 km are concerned, since the rays suffer higher attenuations through higher‐order paths.…”

Propagation of Electromagnetic Waves into Anisotropic Media From an External Point‐Dipole Source

“…the problem is reduced to that of examining the propagation of the component plane waves in a plane-stratified, anisotropic medium. The solution of the latter problem employed here is basically that developed in an earlier paper (Price, 1964), which permits a concise, explicit representation of the solution. Equivalent results can be obtained by using other formulations which have been applied to this problem.…”

“…However, this procedure again restricts the solution to the azimuthally symmetric case, as in the spherical geometry. In the following, the fields are expanded into plane waves, following Weyl (1919;Stratton, 1941), after which full advantage can be taken of the availability of solutions for the propagation of plane waves through arbitrarily anisotropic, stratified media (Clemmow and Heading, 1954;Budden, 1955;Volland, 1962;Johler and Harper, 1962;Heading, 1963;Price, 1964;Pitteway, 1965, Inoue andHorowitz, 1966). Weyl's procedure does not appear to have been previously applied to a source in the presence of anisotropic media.…”

Transmission of VLF Radio Waves Through the Ionosphere

“….With the advent of large, high-speed digital computers, however, tractable numerical methods have been developed for calculating the fields as the wave penetrates into the ionosphere (Clemmow and Heading, 1954;Budden, 1955a, b;Barron and Budden, 1959;Johler and Harper, 1962;Price, 1964;Pitteway, 1965;Inoue and Horowitz, 1966). Although the techniques differ, all have in common the concept of "stratification" or "slabs" in which the ionosphere is figuratively divided into horizontal slabs.…”