1. Introduction

Considerable interest has been shown recently the propagation characteristics of very low quency radio waves [1, 2, 3].2 Electromagnetic. ergy is present in lightning discharges at freencies down to 100 cps and less [4]. There has been a revived interest in very low frequencies LF) for communication purposes because they only slightly affected by ionospheric disturb

for

ranges up to 200 miles or so. VLF has found particularly suitable for navigational ems using a phase-comparison technique [5, 6]. such a system, two continuous-wave signals converted in the receiver to a common comson frequency, which is displayed on a phase er. Lines of equiphase-difference form the ition lines from which this navigational aid, own as "Decca", is based. Another promising plication of VLF is to geophysical prospecting, ere it is known that the propagation of the und wave is dependent on the upper strata of earth's crust. At frequencies of the order of ke, the substrata at depths down to several dred feet will influence the attenuation and se velocity of the ground wave [7].

σ

In view of the present VLF developments in the ve-mentioned fields, it seems very desirable to ent detailed information on the ground-wave aracteristics of an electric dipole source. The quencies considered are 200 cps to 500 kc. conductivity between the source and the server is considered to be essentially homogenewith conductivity a in mhos per meter. The es of a chosen are 4, 10-2, and 10-3, corponding to sea water, well-conducting land, and rly conducting land, respectively. In this quency range the displacement currents in the and can be neglected compared to the conduccurrent, so the dielectric-constant value need be considered. The validity of this approxiation was examined in detail elsewhere [1, 8].

e contribution of the second author consisted of devising the program automatic computer. The responsibility for the technical content of Trests completely on the first author.

ures in brackets indicate the literature references at the end of this paper

The distances in miles from the source to the receiver are taken from 1 to 1,500, inclusive. To account for normal atmospheric refraction [9], the effective earth radius is taken to be k times the actual radius. It should be pointed out, however, that the effective earth-radius concept is not strictly valid at very low radio frequencies. For most of the calculations, k is set equal to 4/3 to conform with standard practice.

2. Some Theoretical Considerations

For any usual source of VLF waves, the equivalent antenna will have a height that is small compared to the wavelength. Furthermore, for most cases the distance of the observer to the source will be large compared to the height of antenna or vertical column of current. In this case, the fields are proportional to the product of the average current I and the height of the antenna. The solution for the field of a vertical dipole of moment Il over a spherical homogeneous earth was first carried out successfully by Watson [10]. Methods for obtaining numerical values for application to radio-wave propagation have been developed by Van der Pol and Bremmer [11], Norton [12], and others. The application of these methods is very straightforward for the purpose of calculating the amplitude of the ground wave for medium and high radiofrequencies. Some special consideration is required, however, at very low radiofrequencies where the induction and static fields of the antenna are not negligible. This is particularly true from the standpoint of the phase variations with distance.

Choosing a spherical coordinate system (r,0,0), the earth's surface is defined by r=a, and the source is a radially oriented electric dipole of moment Il located on the earth at 0=0. A time factor exp(it) is implied throughout. The radial electric-field component E, in the air can be derived from a scalar function II by

where the left-hand side is the ratio of two Hankel functions of the second kind. Van der Pol and Bremmer develop methods for obtaining numerical values of 7, [3, 11].

The "residue series" representation for W is quite suitable for computation when D is large enough for the series to converge rapidly. At shorter distances (down to 1 mile) and very low radio frequencies the series would become impractical, since thousands of terms would be required to secure convergence. At these very short distances, however, the earth can be considered nearly flat and another expression for W(D) is more suitable, which is given by

W=1—i (πp)—2p+i(*)131⁄2 (1—2)

where B=8(1+1/2ẞak).

The three principal terms in the above expressio proportional to 1/D, 1/D2, and 1/D3, can be iden fied with the radiation, induction, and static field respectively. It is clear from the above develo ment that the effect of finite conductivity a earth curvature will affect to some extent all thr components of the field. At very short distanc where pBD, p2828D, and Da are small con pared to one, the expression for the field simplifi considerably to

where W is defined by eq (4) or (6). At inte mediate distances where neither of the above ty special cases is valid, it appears to be necessar in general, to employ the above p series formu for E. For most purposes, however, it is suf ciently accurate to use the following formula f the whole range of D.

where p is Sommerfeld's numerical distance, defined by

3 Private communication.

W.

Dis in the vicinity of unity and for the range of arameters chosen in this report will affect the amplitude to within 1 percent and the phase by ss than 2 degrees. Therefore, eq (11) can be sed for all the calculations.

3. The Numerical Results

The major numerical task in this work was the aluation of the residues series for W. In applitions to field-strength calculation at medium liofrequencies it is usually sufficient to use only ne term of the residue series and then to interpolate with the flat-earth formula, which is valid at short distances. The procedure for making rapid alculations of this type has been developed by Norton [12]. For amplitude and phase calculaons at VLF, the interpolation method is not of fficient accuracy, and more terms of the residue eries are required. In fact, it was considered

TABLE 1. Amplitude of W (k=4/3)

Distance (miles)

calculations from the residues series formula at distances down to 37 miles, where no interpolation with the series formula is required. In fact, there was a region of overlap in the calculations that provided a convenient check on the computations. Due to the excessively large number (up to 1,000) of terms required to secure convergence of the residue series at short distances and frequencies as low as 200 cps, it was summed on the SEAC (the National Bureau of Standards electronic automatic computer). Values of the amplitude and phase of Ware provided in the tables 1 and 2, respectively, for several conductivity values and various distances. There are several gaps in the tables that correspond to situations where it was not possible to devise a convenient program. In some cases this occurred where the Bremmer-type [3] series formulas for & did not converge.

TABLE 2. Phase of W (k=4/3)

(The values are all negative)

Distance (miles)

are of considerable intrinsic theoretical interest, since they can be used to verify the validity of operational methods used to calculate the field of a dipole near gently curved surfaces. The p series formula for I was derived by such a method.

Using eq (8) and (11) in conjunction with the tabulated values of W, the amplitude E and phase of the vertical electric field are plotted in figures 1 to 16. The frequencies chosen are 0.2, 0.5, 1.0, 2, 5, 10, 20, 50, 100, 200, and 500 kc. The groundconductivity values are 4, 10-2, and 10-3 mho/m, corresponding to sea water, well-conducting land, and poorly conducting land, respectively. The values shown in these curves can be compared directly with some earlier calculated results [1], using essentially Norton's method [12] for the frequency range 15 to 500 kc. The agreement is usually within 2 percent for the cases that could be compared directly. It appeared that the maximum discrepancies occurred in regions where the earlier curves were interpolated between the flatearth values and the first term of the residue series. As mentioned above, no interpolation is required in the present curves, so they are believed to be

more accurate.

To conform with standard practice [9], the ratio k of the effective to the actual radius of the earth is taken to be 4/3 in the calculations. To illustrate the effect of modifying the earth's radius, factor k from 1.0 to 4/3 is illustrated in figures 17 and 18. The amplitude E and the phase lag o are plotted as a function of frequency for distances of 124 and 621 miles with a ground conductivity of 10-2 mho/ m. The difference between the two sets of curves for k-1 and 4/3 is not insignificant, particularly from the standpoint of phase. However, at distances less than 150 miles or so, and at frequencies less than 100 ke, the amplitude difference is within 1 percent, and the phase difference is less than 2 degrees.

4. Conclusion

It is believed that the curves presented in this report will provide basic information on groundwave propagation in the low, very low, and ultralow frequency bands. It should be emphasized that the ionosphere has not been considered here,

and consequently the total field at larger ranges

ever, by using special techniques, to eliminate reduce the "sky wave" even at large distances & that the computed values of the ground-way propagated fields should have some significant even out to 1,000 miles.

We thank Loris Perry for assistance with th calculations, and A. G. Jean and J. R. Johl for their helpful suggestions.

5. References

[1] J. R. Wait and L. L. Campbell, Transmission curv for ground wave propagation at V. L. F., Defen Research Telecommunications Establishment R port (no. R. 1), Ottawa, Canada (1952). [2] K. G. Budden, The Propagation of low frequent waves to great distances, Phil. Mag. 44, 504–51 (1953).

[3] H.

Bremmer, Terrestial radio waves (Elsevi Publishing Co., New York, N. Y., 1949). [4] H. Norinder, Variations of the electric field in th vicinity of lightning discharges, Arkiv. Geopysi Bd. 2, No. 20, Stockholm (1952)

[5] A. B. Schneider, Phase variations with range of t ground wave from C. W. transmitters in the 70 130 ke/s band, J. British Inst. Radio Engrs. 1 181-191 (1952).

[6] B. G. Pressey, G. E. Ashwell, and C. S. Fowler, T measurement of the phase velocity of ground way propagation at V. L. F. over a land path, Pro Inst. Elec. Engrs. 100, pt. III, 73-84 (1953). [7] J. R. Wait, Theory of electromagnetic surface wav over geological conductors, Geofisica Pura e App cata 28, 47-56 (1954).

[8] J. R. Wait and L. L. Campbell, Effect of a lar dielectric constant on ground wave propagatio Can. J. Phys. 31, 456-457 (1953).

[9] S. S. Atwood and C. R. Burrows, Radio wave prop gation, OSRD, Summ. Tech. Report, Committ on Propagation (1946).

[10] G. N. Watson, The diffraction of radio waves by t earth. Proc. Royal Soc. (London) [A] 95, 546-5 (1919).

[11] B. Van der Pol and H. Bremmer, The diffraction electromagnetic waves from an electrical poi source round a finitely conducting sphere. P Mag. 24, 141–175 and 825–864 (1937); 25, 817-8 (1937); 27, 261-275 (1939).

[12] K. A. Norton, Calculation of the ground wave fi intensity, Proc. Inst. Radio Engrs. 29, 623-6 (1941).

[13] J. R. Wait, Radiation from a vertical antenna ove curved stratified earth, J. Research NBS (1956) RP2671.