A flexible theoretical model plas ma which can be deformed to fi t most mea s ured elect ron-ion-altitude profiles i. 3 e mployed t ogether wi th a va ila ble geophysical dat a on t he ionosphere to evahlate reflections and trans missions durin g quiescen t and disturbed propagation conditions. The refl ect ions and t ra ns mi ssions in t he ionospher e are determin ed rigorously with t he aid of th e classica l mag neto-ionic t heory. The complex indexes of r efr ac tion of t he medium are dedu ced, and a coupling in t he plas ma betwee n ordinar y a nd extraordinary, upgoing a nd down going mode of propagation is investigat ed. Th e corres ponding refl ection a nd t ransmi ssio n coeffici en ts are then ca lculated, a nd cer tain phenomena which can be exp ected as a r es ult o f the action of a solar di sturbance on t he r eflection process are predicted. Th e di sturbance of solar origin, investigated as a n application of developed t echniques, influences the refl ected and trans mitted LF wa ves in th e lower ionosph ere in a complicat ed manner. Howe ver, the high ab orption phenom ena exhibited by high fr equ encies do not see m t o exist for t he plas ma profiles investigated with the classical magneto-ionic theory. Th e electron-collision frequ e nci es of the classical magnetic-ionic theory ar e modifi ed to introduce a collis ion fr equ ency proportional to the electron energy, and the changes necessary in th e formulfttion of the classical theory as a r es ult of such a consid eration arc presented.

A technique is described for determining the conductivity of the ground at low frequencies with the aid of sferic pulses from thunderstorms. The paper is illustrated with an actual conductivity determination, and the detailed comparison of waveforms predicted theoretically with those observed experimentally indicates that an effective value of conductivity can be measured to a precision of one or two significant figures. The results of the analysis indicate application of both experimental and analytic techniques to other propagation studies, such as the evaluation of the reflection and transmission properties of the lower ionosphere.

The rigo ro us appli cation of t he magn eto-ionic t heory to t he calculation of re flection coeffi cients for a sha rpl y bound ed ionospher e mod el is carri ed ou t . The p ap er is illu str ated with compu tations applicable to the D-region or t he E-region of t he ionosp her e. The quasilongi t udinal approxim ation is d eriv ed from this t heory a nd the r a nge of validi ty of this a pproximatio n is illustra ted . The r estri ctions imposed by t he use of a s ha rply bounded mod el ionospher e a re discussed .

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The rigorous mat he mat ical t reatment for the propagation of a radio wave from a H ertzdipole-sou rce curre nt-moment a round a finitely condu cting sphe ri cal earth surrounded b y a con centric electron-ion plasm a can be expressed as a series of zonal harmonics. S uch a solution to the prob le m was obtained for t he terrestrial sphere without a conce nt ric plasma many years ago (190'1-1915). Howeve r, the summ ation of t he seri es, eve n at long wavelen gths o r low fre quencies, was considered to be impractical and t he well-known a nd, indeed , rigorous \' Vatson transformation was introduced (1918). The •Watson transfo rm ation led to the development of elegan t mathematical techn iques both rigorous and app roximate for t he evaluation of the fields of r a dio waves in t he vicinity of t he ear th. However , it does not n ecessarily follow t hat the Watson t ra nsform ation is t he only way to achieve numerical master y of t he problem. Indeed, i t also does not fo llow t hat the Watson transformation is t he most effici en t approach to the rigorous form of the theory of propagation , especia lly at long wavelengths. This p a per demonstrates that t he field of t he propagated long wavelength radio wave (fre quenci es less than approximately 50 kc/s) can indeed be evaluated by a summation of a seri es of zonal harmonics. Whereas t he Dumber of terms could become quite large (of t he order of 10 k[a wher e a is the radius of t he sphere a nd k[ is the wave number of the m edium between t he concentric plasma and t he earth), the speed with which these terms can be summed on a large-scale co mputer offsets the complicatio ns introduced by the Watson t ransformation as to the rigoro us m at hem atical solu t ion of the problem. The detailed structure of the field in the a bse nce of a concentric plasma is characterized b y t he quite r eg u lar behavior of the ground wave as a func tion of distan ce. Indeed, the steady decrement of t he ground-wave field is modified on ly near the a ntipode, where an interference pattern or standing wave as a fun ction of distance is no te d because of a nother wave's traveling a round t he sphere in the opposite direction. The introduction of t he concentric electron-ion plasma shell traps t he waves lcaking into space, whem reflection frOlu. the plasm a builds up t raveling waves in the direction of increased distance from t he transmitter. Thus, the series of zo na l hannomi cs co mprises individual waves which a re traveling in the radial direction with r es pect to t he center of the sphere and standing in t he direction of increased angular distance around the spher e. These waves, when summed, build up the wave progressing in the direction of increased angu lar distance. Under special circumstances, standing waves can be noted. This is especially obvious near the antipode of the transmitter. The results of the computations indicate that full rigor can be achieved with comparative ease at fre quencies less t h an approximately 50 kc/s, leaving on ly t he assumed m...