The value of the phase, c, of the secondary factor, F, is illustrated graphically in figure 2. The phase, e, for the free space field secondary factor, F, as shown was computed from eq (24). The value of the phase, c, due to the disturbing influence of the earth is also shown at a frequency of 100 kc for conductivities of 5, 0.005, and 0.0001 mho per meter and for plane and spherical earth theory computations of the secondary factor, F.

The details of the computation of the secondary factor, F, or the Hertz vector, II, near the earth have been elegantly worked out by the aforementioned authors. The vertical electric field intensity at great distance close to the earth in the spherical-earth theory is, after the various approximations [6, 7], as follows: 13

Close to the source (wavelengths), the effect of the source on the secondary factor, F, must be considered. In this case, at the surface of the earth, i. e., fs(h2)=fs(h1)=1:

Because the computation by means of the residue series described in eq (28) becomes unwieldy at short distances, it can be assumed that the earth is a plane. In the plane-earth theory, the vertical electric field intensity at the surface of the earth, E2, was found to be given by the following approximate formula [2, 3]:

It has been found convenient to express the phase, o, as a time, t, as follows:

For the phase, c, of the secondary factor, F, a time t, can be ascribed as follows:

(34)

(35)

The vertical lapse of the permittivity of the earth's atmosphere was reckoned from a mathematical model of the atmosphere which assumed a linear decrease with altitude from some surface value 7, eq (12), and some value, ne, h kilometers aloft. This is expressed as AN units per kilometer as follows:16

where a is the radius of the earth, a, is the "effective radius" of the earth, and k' is the "effective radius factor" of the earth. AN is related to the parameter, a, in eq (26), (27), and (28) as follows:

A physical interpretation of the a parameter results in an adjustment of the amplitude and phase of the ground wave which would be expected from the assumed mathematical model of the atmosphere, i. e., the assumed vertical lapse of the permittivity of the atmosphere. The derivation of this parameter by Bremmer [7] is based on a model of the earth's atmosphere which has a linear decrease with altitude. The derivation is completely independent of ad hoc geometrical-optical considerations with which it is often associated. It is independent of the thickness of the earth's atmosphere (wavelengths) as long as a wavelength is small compared with the radius of the earth. Satisfactory experimental vertification of this model has not as yet been achieved. Other mathematical models of the atmosphere could put further restrictions on this parameter, especially at low and very low frequencies.

3. Results of the Computation

The results of the computation are tabulated in tables 1 to 44, and are presented graphically in figures 2 to 17.

The phase, e, of the secondary factor, F, in free space, (=0), eq (24), is shown in figure 2. This is also the phase of the secondary factor over an infinitely conducting earth (=). The disturbing influence on the phase, oc, of the finite conductivity of the earth is also shown at a frequency of 100 kc, for three values of conductivity: σ=5, 0.005, and 0.0001 mho per meter for both the plane- and the spherical-earth theory.

15 See figure 16.

16 See figure 10.

surface of the earth from the source, conductivity of the earth, and the vertical lapse of the permittivity of the atmosphere on both the phase of the secondary factor, oc, and the amplitude, E, of the ground wave. It is of computational interest to note that the value of the phase. , and the value of the amplitude, E, computed from the plane-earth theory and the sphericalearth theory at moderately decreasing distance, approach each other such that at short distances the two computations become identical. This is illustrated for the phase, c, in figure 3. If required, the total phase may be computed quite simply from eq (16). The amplitude of the ground wave, E, volts per meter, assumes that the source dipole momentum, Il, eq (22), (26), and (30), has a value of unity (1 ampere-meter). The effect of the antipode, or the effect of the wave progressing around the earth in the opposite direction (- direction in fig. 1), was found to be negligible as far from the source as 10,000 miles. Also, note that the value of the attenuation is very large for the ground wave (figs. 11, 12, and 13) at such distances from the source. The variation of the phase, c, of the secondary factor, F, with the parameter a is shown in figure 4. The corresponding amplitudes, E, are shown in figure 13. The curves, a=1, correspond to a value that would be expected from a homogeneous atmosphere. However, the average atmosphere is not homogeneous, due in large measure to the gravitational field of the earth. The value, a=0.75, is a value frequently assumed and corresponds to an average vertical permittivity lapse such that the "effective radius factor," k', eq (38), has a value of 4/3. On the other hand, the value a=0 corresponds to the plane-earth theory value. Such a vertical lapse or greater is often considered at very high frequencies to account for the effect of ducting. Further graphical study of this parameter is shown in figures 6 and

10.

Considerable interest has developed in the effect of conductivity and conductivity changes on both the amplitude and phase. The computations of this report assume homogeneous conductivity. The effects of conductivity on the amplitude and phase are shown in figures 3,7, and 12. Of particular interest is the phase, pe, of the secondary factor, F, shown in figure 7 for various distances for both the plane- and spherical-earth theory. Between the asymptotes =0 and σ a maximum phase retardation, e, is observed. This is in the region of maximum energy absorption by the earth and includes the values of conductivity most frequently observed on the earth. The asymptote, σ, is quite rigorous for a highly conductive sphere the size of the earth. However, the value σ=0 does not represent a dielectric sphere as the terms of the rainbow series So, S1, S2 ..., St (for k>-1) described by Bremmer [6,7] are neglected. This approximation was considered quite reasonable for finite or infinite values of conductivity, σ, as the rainbow terms other than S-1 are very highly absorbed by the earth.

The amplitude of the ground wave E is slightly enhanced as the value of the conductivity, σ, is increased through values of o=0.05, σ=5 (sea water), and σ=∞. This is due to the behavior of the attenuation factors, exp[-Im 7.] [7]. The value of Im 7, for the first term of the residue series (eq (26), s=0), which in large measure determines the attenuation is shown graphically in figure 17. A similar curve for the phase parameter, Re 70, is shown in figure 19.

4. Physical Significance of the Phase

At a time, t, and a distance, d, from the source, the signal, f(d,t), upon which a measurement is made may be described as follows [28, 29]:

f(d,t)=fo(d,t)+f1(d,t)+f2(d,t).

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In this paper, the continuous wave signal, fo(d,t) has been evaluated. This continuous wave signal will, in general, be altered by the transient terms, fi(d,t) and f(d,t). The measured phase will not in general correspond to the values computed from fo(d,t). The transient terms are usually evaluated for specific systems of measurements under study and are considered beyond the scope of this paper. The value of the phase of the continuous signal, fo(d,t), may, however, under carefully considered conditions have physical significance.

be interpreted as the time required for the signal to propagate to the observer. The value of the phase, te, in this paper, is such a time relative to the free-space propagation time. This measurement can also be described as a velocity measurement if the distance from the source is considered. It has been shown by Sommerfeld [28] and Brillouin [29] that the signal velocity, , is always less than the universal constant, c. At great distance from a source in free space. the signal velocity, , corresponds to the phase velocity, te, or the group velocity, . More often, however, the signal velocity and signal time-delay correspond to the group. However, under certain conditions neither the group velocity nor the phase velocity have physical sig nificance. The phase velocity and the group velocity have been implied in this computation and may be explicitly developed.

If the time-dependent factor is included, the phase, 4, of the ground wave is as follows:

The group delay can be estimated by Kelvin's principle of stationary phase [25] as follows:

The velocity of propagation, e, or the phase velocity, v., is as follows:

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(44)

(45)

the velocity of the group, t, may be computed from eq (24) as follows:

At intermediate distances from the source, the group delay, group velocity, and phase velocity may be evaluated by graphical construction of the factors:

At great distance from the source, close to the earth, the group velocity and group delay may be estimated from the first term of the residue series as follows: