A method for obtaining the wave solutions of ionospherically reflected long waves, including all variables and their height variation

A method is presented for obtaining solutions to the one-dimensional wave equation which arises in the course of the application of the wave theory to the study of the ionosphere. The π's represent characteristic wave functions of the ionosphere, K0 the propagation constant of free space, ∈(x) the complex index of refraction of the ionosphere, and the primes denote differentiations with respect to the space coordinate x. The method has been applied to the following “E-layer” problem with good results. Operating frequency —150 kc/sec Angle of incidence —NormalN (electron density) —Chapman spatial distribution with various values of maximum electron density ν (collisional frequency) —Exponentially decreasing with heightH (scale height) —10 km (constant) Earth's magnetic field —Evaluated at 100 km over State College, Pennsylvania Dispersion theory —Appleton-Hartree (Sellmeyer) An example of the method for obtaining the wave functions and reflection coefficients is carried out in detail, and reflection coefficients are presented for five different values of maximum electron density. This method is compared with the ray optics and the W.K.B. method. The results are compared very briefly with experiment.