Up-going Waves Research Articles

Some properties of spheroidal modes of a homogeneous elastic sphere with special reference to radial dependence of displacement

A ray-theoretical approach is attempted to interpret radial eigenfunctions (radial and tangential displacements) of spheroidal modes of a homogeneous elastic sphere. It is shown that surface displacements of the mode solutions agree well with those expected when relevant body waves are incident on the free surface. The possibility of interpreting amplitude dependence on depth in terms of interference phenomenon of up-going and down-going body waves is also demonstrated. The radial eigenfunctions are computed for a variety of spheroidal modes with relatively high radial mode numbers. An interesting feature is found in these functions: The number of node surfaces in the radial distribution of the radial displacement increases systematically for a given radial mode in conformity with a change of surface value of the tangential displacement from negative to positive with increasing colatitudinal order number.

The second Z-propagation window

ELLIS1,2investigated the condition in the ionosphere under which an upgoing ordinary radio wave is converted, through a coupling process, into an upgoing extraordinary wave in the Z mode. (Propagation in the Z mode requires the presence of both an ionised gas and a magnetic field; hence such waves cannot be received in free space.) This occurs for a wave incident on the ionosphere from below with its wave normal close to a critical direction in the magnetic meridian plane. This direction is such that when the wave reaches the level where X = 1, which is normally a complete reflector for the ordinary wave, its wave-normal direction has become close to the magnetic field line direction. The symbols X and Y are as usually defined in magnetoionic theory (see Ratcliffe3). The level X = 1 becomes semi-transparent for a cone of angles centred on the critical direction. This cone is called the ‘Ellis window’. The Z-mode waves are reflected at a higher level where X = 1+Y. Here I wish to draw attention to a window which is relevant to the escape of Z-mode radiation produced naturally in the topside ionosphere

Numerical Solution of the “Ellis Window” Problem

THE technique of complex ray tracing for studying problems of radio wave propagation in the ionosphere has recently been put forward1,2. A particular application of this technique is the determination of the angular diameter of the “Ellis window”. Ellis3, 4 investigated the conditions under which an upgoing ordinary wave is converted, through a coupling process, into an upgoing extraordinary wave in the Z mode. This occurs for a radio wave incident on the ionosphere from below with its wave normal close to a critical direction which is in the magnetic meridian plane at an angle θc to the vertical. Here sin θcl{Y/(Y+1)}1/2 where l is the sine of the angle Θ between the Earth's magnetic field and the vertical and Y and X are the standard symbols of magneto-ionic theory (see Ratcliffe5). A level near X=1, or just below, in the ionosphere is normally a complete reflector for the ordinary wave, but is semitransparent for a cone of angles centred on the critical direction. This cone is called the “Ellis window”. Its diameter is defined to be the range of wave normal angles within which more than half the incident energy flux is transmitted through this level.

Magneto-ionic multiple splitting determined with the method of phase integration

The method used is a first-order W.K.B. approximation with an integration path in the complex height plane. The refractive index versus height is represented on a Riemann surface with four corresponding to upgoing and downcoming waves and ordinary and extraordinary types of polarization. The branch points connecting the streets are characterized as reflection-points or ascoupling points. Suitable paths of integration in the Riemann surface determine the reflection coefficient for the various magneto-ionic components. Five possible fundamental components have been found, and an infinite number of additional components due mainly to multiple reflections between the different reflection points. Numerical computations have been carried out for ionospheric model with a Chapman distribution of electrons and an exponential decrease of collision frequency located in a moderate magnetic latitude. The results are represented in a sweep frequency picture of virtual height and absorption for five fundamental modes of propagation.

Wave solutions, including coupling, of ionospherically, reflected long radio waves for a particularE-region model

This paper is concerned with the solution of the coupled wave equations which arise in the application of the wave theory to the problem of ionospheric wave propagation. It concerns an extension of previous work [see 1 of “References” at end of paper], which will henceforth be referred to as [1]. The method of “variation of parameters” is used to obtain approximate solutions to the coupled equation from the uncoupled solutions obtained in [1]. A diurnal and seasonal model representing the E-region of the ionosphere above State College, Pennsylvania, is considered. This model, as in [1], consists of a Chapman-like E-region whose maximum in electron density is at a constant height. Approximate 150 kc/sec wave solutions including coupling are obtained for this model. These wave solutions, of course, exhibit the well-known reflection condition corresponding to an electron density of around 3,000 electrons/cm3. It is shown that the effect of the coupling is to cause a wave traversing a coupling region to excite a new wave propagated in the direction of propagation of the incident wave and also a back-scattered wave propagated in the reverse direction. As indicated below, these coupling effects become important in our model below the “reflection” level. The back-scattered wave due to the initial upgoing wave will thus appear as a reflected wave originating in the coupling region. The forward-scattered wave due to the down-going wave from the upper “reflection” level also must be considered, particularly in calculating the polarization of ionospherically reflected waves. It is shown that, in the case of 150 kc/sec waves, the coupling effects occur in the neighborhood of N = 300 electrons/cm3, which corresponds to the “classical reflection” level for the “ordinary” wave. The coupling effects become greater as the collisional frequency, ν, associated with the coupling N value decreased toward the critical collisional frequency, νc [2], for the night-time models due to the increase in heights of the “bottom” of the layer at these times. This results in stronger split echoes and greater departure from circularity in the polarization ellipses. The results of computations utilizing the theory, in comparison with experimental results, indicate that the stratification observed in the height records may be explained as outlined above. The absorption and polarization experimental results, however, cannot be accounted for by the E-region model utilized alone. The consideration of an electronic D-region, however, readily accounts for all three experimental factors, as will be discussed in detail in a later paper.