J. R. Johler, W. J. Kellar, and L. C. Walters The special theoretical considerations pertinent to the computation of the phase of the ground wave at low radiofrequencies are discussed. The formulas necessary for the numerical evaluation of the amplitude and phase, and the results of the numerical computation are presented. The effects of frequency, conductivity of the earth, altitude above the surface of the earth, and vertical lapse of the permittivity of the atmosphere are evaluated. 1. Introduction The phase of the low-frequency ground wave is implicit in its theoretical development. Most computations in the past have considered only the amplitude. However, the importance of a precise computation of the phase has increased with the use of various systems of lowfrequency radio navigation [21, 22, 23, 26, 27]. The phase of the propagated ground wave is also an important consideration in the solution of the more general problem of the propagation of a transient in a radio-communication system. Computations have been performed by the authors during the past 2 years for the purpose of evaluating the effects of propagation on various radio-communication systems and are presented here in a unified form. A mathematical model for the computation of both the amplitude and the phase of the ground wave near the earth has been presented by Sommerfeld [11], Wise [9, 10], Van der Pol and Niessen [15], Watson [8], Van der Pol and Bremmer [6], Bremmer [7] and Norton [1, 2, 3, 4, 5]. This Circular concerns the computation of the phase of the low-frequency ground wave from this mathematical model. The amplitudes of the ground waves incident to this computation are also presented. The phase variation with frequency is considered. The disturbing influence of the source is considered for the particular source model used in this computation. The effect of the finite conductivity of the earth and the vertical lapse of the permittivity of the atmosphere is also demonstrated by this computation. 2. Theory of the Phase Computation The prime consideration in this study of the propagation of the ground wave was the nature of the source. The source selected for this computation was the elemental vertical electric dipole of the type originally proposed by Hertz [20] and which is used for the ground wave in various modified forms by Norton [1] and Bremmer [7]. The field developed by the source was described by a primitive quantity, II, (volts meters, or joulesmeters/coulombs) known as the Hertz vector [20]. The various electric and magnetic components of the field can be computed from this quantity by a differentiation process. As the field can be described by a single quantity, II, the Maxwell theory for the problem at hand can be represented at a primitive source, , as follows: 3 The work represented by this paper was sponsored by Wright Air Development Center, Wright-Patterson Air Force Base, Dayton, Ohio, in connection with Air Force Contract 33(616)-54-7. 'Figures in brackets indicate the literature references on page 10. The rationalized mks system of units is used throughout this Circular. The Hertz vector for the solution of this equation is as follows [19, 24]: (2) (3a) (3b) (3c) The evaluation of the source at a time, t— (nd)/c, instead of a time, t, accounts for the finite propagation time through space. As the computations in this paper were built upon two separately derived theories (plane- and spherical-earth theories), the source current, i, must be specified closely if consistency is to be maintained at intermediate distances from the source. Hence, the source, i, was assumed (a) to have a uniform distribution along an elemental linear distance, l, of the z axis of the coordinate system 5 (at the origin in cylindrical coordinates; at the surface of the earth, r=a, 0-0 in spherical coordinates), and (b) to vary sinusoidally with time at an angular frequency w. It should be noted that the length, 7, would represent the effective length of a practical low-frequency radio-transmitting antenna. The effective length would equal half the physical length if the current on the antenna had a linear distribution along its length. The distance, d, is the distance from the observer to the volume element of integration in eq (3). The distance, d, is further specified as the distance measured along the surface of the earth. The source polarization, , in the spherical-earth theory was then specified as follows: The source was therefore always considered to be polarized perpendicular to the surface of the earth. In either case, however, The nature of the oscillator supplying the source with electric power may be described as follows: Rei Io sin (wt+). (5) It is quite evident that due to the assumptions in eq (4), the phase of the oscillator at time t=0 was advanced radians. This specification simplified graphical presentation of the results of the computation and also avoided negative phase values for the "secondary factor" to be defined later. It is also evident that the amplitude of the oscillator does not vary other than sinusoidally with frequency w The volume element of integration, eq (3), now becomes a linear element of integration. See figure 1. See figure 1. 7 Vectors are defined in figure 1. The amplitude of the field is inversely proportional to the frequency, w, and the distance, d, and therefore becomes infinite when either w or d is zero. With the aid of this representation of the field, the electric and magnetic components of the field can be computed as follows: E=curl curl II (volts per meter) k2 (7) (8) The parameter k is defined for a medium of permittivity, «, farads per meter; permeability, 40, henrys per meter; and conductivity, o, mhos per meter as follows [18]: 8 The scalar electric and magnetic components of the field can now be computed. The vertical electric field intensity, E, or E, is of particular interest." One of the most important and difficult theoretical problems with which early workers in the field of radio-wave propagation were confronted was the evaluation of the Hertz vector II near the surface of the earth. The results of the efforts of the many workers in the field are presented in form suitable for computation by Van der Pol and Bremmer [6], Bremmer [7], and Norton [1]. If the Hertz vector, II, is evaluated for the effect of the proximity of the earth, the ground wave for purposes of this study is now completely specified. However, it has been found convenient to express the total field as the product of two factors: (a) the primary field or free-space field, Epr (volts per meter); and (b) the secondary factor, F,10 (dimensionless). The primary field is defined as follows: The wave number, k1, refers to the air medium at the surface of the earth. E F= 2Epr More detailed formulas are given in appendix I. Other components are given in appendix I. 10 F is equivalent to the ratio of Hertz vectors II/2I1p, in the spherical-earth theory described by Bremmer [7, p. 48]. of the source or the earth. At very great distance from the source, in free space, σ=0, F= 1; or over perfectly conducting earth, σ, F=1. It is of interest to note, however, that a secondary factor other than unity defined by eq (13) exists over perfectly conducting earth close to the source, or other than in free space close to the source due to the disturbing influence of the source. This consideration at low frequencies is of considerable importance, because "close to the source" can be as far away as several hundred miles. Wait [16] has shown the effect of the horizontal stratification of the ground on the secondary factor, F. This paper is concerned with the homogeneous case only. At great distances along the surface of the earth, the phase, c, of the secondary factor, F, was computed from the spherical-earth theory. Close to the source of the radiation, the computation by this method became unwieldy. The phase was then computed from the planeearth theory. The computation of the phase, p', of the primary field as defined by eq (12) was quite simple, and, neglecting the time function, wt, is C or is the phase of the secondary factor, F. At great distance from the source, d' is a large number (thousands of radians). On the other hand, ø, is a relatively small number (between 0.1 and 10 radians), and can therefore be regarded as a phase correction that is to be added to the free space or primary field to account for the disturbing influence of the source or the earth. The basic assumptions and specifications described were so defined as to avoid a negative phase, c, for the secondary factor, F, thus simplifying the graphical constructions of the functions. In this regard, the free-space wave field intensity close to the source of the radiation is of interest and may be described as follows:"1 |