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...