NEW YORK UNIVERSITY INSTITUTE OF MA', L SCIENCES LIBRARY SI &***** W**, N«w York 3, N. Y. ^E ET PR/J. ^\ tx r% NEW YORK UNIVERSITY iWl > t* Institute of Mathematical Sciences tri su ^-^ j> Division of Electromagnetic Research RESEARCH REPORT No. EM-79 Long-Range Propagation of Low-Frequency Radio Waves between the Earth and the Ionosphere J . SHMOYS CONTRACT NO. A F- 1 9( 1 2 2 )-42 MAY 1955 NEW YORK UNIVERSITY Institute of Mathematical Sciences Division of Electromagnetic Research Research Report No. EM-79 LONG-RANGE PROPAGATION OF LOW- FREQUENCY RADIO WAVES BETWEEN THE EARTH AND THE IONOSPHERE J. Shmoys h Shmoys Morris Kline Project Director The research reported in this document has been made possible through support and sponsorship extended by the Air Force Cambridge Research Center, under Contract No. AF-19(122)-42. It is published for technical information and does not necessarily represent recommendations or conclusions of the sponsorship agency. New York, 1955 - i - Abstract The problem of modes of propagation of electromagnetic waves between a perfectly conducting earth and a gradually varying ionosphere is considered. The case of exponentially varying ionospheric parameters is solved in terms of Bessel functions. The propagation constant, the angle of arrival and the group velocity are calculated for the first few modes of propagation. It is shown that the results for the phase and group velocities and angle of arrival of low-order modes obtained when the ionosphere is assumed to be a perfectly con- ducting sheet at a height simply related to ionospheric parameters are very close to the true values. Table of Contents Page 1. Introduction 1 2. Formulation of the eigenvalue problem U 3. Lossless ionosphere. Horizontal polarization 9 U. Lossless ionosphere. Vertical polarization 12 5. Lossy ionosphere. Horizontal polarization 13 6. Lossy ionosphere. Vertical polarization 22 7. Applications 23 References 28 - 1 - 1. Introduction The problem considered here is that of propagation of low-frequency radio waves (frequency under 15>0 kc.) along the surface of the earth. This problem was first treated by Watson'- -* who assumed that the ionosphere is a homogeneous isotropic medium of large but constant conductivity, occupying all space above a given height over the earth's surface. In much of the sub- sequent work on this problem, the assumption of a sharp discontinuity between the atmosphere and the ionosphere was retained in the model used. *- ' ' ' ' ■» . Rydbeck'- , however, considered continuous models with discontinuity in first or second derivative of the conductivity and calculated the attenuation coe- fficient approximately. We shall consider the ionosphere to be an isotropic medium with conductivity varying exponentially with height. Instead of considering the problem in spherical geometry, we shall demonstrate that the error incurred by treating the earth as a plane and the ionosphere as a plane stratified medium is negligible for the parameters we shall calculate. There are two basic approaches to the theory of propagation of radio waves between the earth and the ionosphere. One is the mode theory of Watson, also used here. The other, far simpler theory, commonly used in short wave propagation problems, is that of geometrical optics, i.e., one assumes a zig- zag path connecting the receiver and transmitter. We should point out that the two theories give different results for the quantities we wish to calculate. Referring to Fig. 1, we see that in the ray theory the angle of arrival would be a function of the distance between the receiver and the transmitter, while in the wave theory it is not. Also, in the ray theory the time of travel in a constant number of hops from T to R is not proportional to the distance TR, while in the wave theory that ratio is constant. - 2 - Fig. I Zig-zag path T- transmitter R- receiver Since we are dealing with very low frequencies and propagation over large distances, we must use the wave theory. We are concerned here not with the detailed structure of the elec- tromagnetic field but with certain parameters characterizing the various modes of propagation, namely group velocity, angle of arrival, and attenuation. We show that in the case of exponential variation of electron density and collision frequency we can represent the ionosphere, for the most practical purposes, by a conducting sheet at a height simply related to the physical parameters of the ionosphere. The results obtained using this model is consistent with experi- mental results in the propagation of certain low-frequency atmospherics which have been called *tweeks'» We start from the scalar wave equation satisfied by a single-component Hertz vector. By separation of variables in spherical coordinates we reduce - 3 - the problem to that of finding the eigenvalues of a certain second- order differential operator. Next we demonstrate that the eigenvalues thus obtained differ only by a negligible amount from the corresponding eigenvalues obtained for a flat earth. Having thus formulated the eigenvalue problem we consider first the case of the lossless ionosphere. Horizontally polarized waves are shown to have certain properties independent of the actual variation of electron den- sity with height. The case of vertically polarized waves in a lossless iono- sphere presents difficulties which will be pointed out. Next we consider the problem of propagation of horizontally polarized waves in a very lossy ionosphere. In this case we also have an infinite set of modes, although these can no longer be sharply divided into propagated and attenuated ones. The first few modes are, in general, little attenuated- higher- order modes are heavily attenuated. Assuming exponential variation of electron density and collision frequency, we can calculate the eigenvalues of the first few modes. The group velocity and angle of arrival thus calculated are shown to be consistent with those obtained for a simpler model of the ionosphere - one in which the ionosphere is replaced by a perfectly conducting sheet at some height above the earth. This effective height is calculated in terms of ionospheric parameters. In the case of vertical polarization in a lossy ionosphere there is again an additional difficulty which makes the theory applicable only to fre- quencies above ca. 2 5 kc» Finally the application of our model to the problem of propagation of low-frequency atmospherics, 'tweeks', will be discussed. - u- 2« Formulation of the eigenvalue problem The electric and magnetic fields in spherical geometry can be derived from a single component Hertz vector. Thus the vector problem is reduced to a scalar one. The equation to be solved is: (1) 2 * v^ TT + k • TT - o, where e is related to the complex dielectric constant as follows 1 ' (i) in the transverse electric case (E^ « E ■ Hj * 0) (2) * 6 - &} (ii) in the transverse magnetic case (Hv a H - E^ ■ 0) (3) ♦ ,-2 1/2 / -1/2 v e = e - k e ' (e ' ) . v rr Xl9» ,°) Fig. 2 Spherical geometry - 5- The Hertz vector must vanish on the surface of the perfectly conducting earth (r = a) in case (i) while its normal derivative must vanish in case (ii). At infinity, the Sommerfeld condition is to be applied. The dielectric constant of an ionized medium of electron density N per cubic meter and collision frequency v is w « ■ 1 " =s&7 ' Y " 3>19x10 " 3 ' where co is the angular frequency of the field. When co is much smaller than v, one can approximate e by (5) e ** 1 ♦ 3? • In fact, for low frequencies, oo is negligible with respect to v in the D layer and in a large portion of the E layer. Furthermore ( since yN/cov is much larger than one in most of the ionosphere, we can consider the dielectric constant in those regions as purely imaginary. This is equivalent to saying that the iono- sphere behaves very nearly like a conducting medium, with conductivity o", (6) — P + — -x + k e (x) 3x^ by IT- o obtained by separation of variables, we would have obtained the relation (19) 2 -2 (0 + ia) c ■ v(v+l)a • - 9 - p The relative error in (B+ia) is then of the order of l/8v , or, since v is of the order of ka, l/8(ka) • This error is entirely negligible, since even for a wavelength of 100 km it amounts to one part of per million. We may therefore drop the spherical geometry of the original problem and consider plane wave solu- tions of equation (18). Then we set (20) TT - u(x) e i(P+ia)y - u(x) e^ y , so that u satisfies the equation (21) u^ ♦ [k 2 e*(x) ♦ T 2 ] u - 0. The eigenvalue parameter \ has a discrete spectrum for the class of functions 6 (x) considered here. The boundary conditions are different in the two polar- izations, namely: (i) u(0) ■ for horizontal polarization (TE case) (22) (ii) u (0) - for vertical polarization (TM case). Because of the behavior of e at infinity we may require u to be bounded or even square integrable. 3. Lossless ionosphere. Horizontal polarization In the case of horizontally polarized waves propagated in a lossless ionosphere, we have, setting v - in fiq. (U): (23) 6* - 6 - 1 - YNoT 2 - - 1 - Yc" 2 Nk" 2 . - 10 - Substituting this expression in the differential equation (21), we obtain (2U) U xx + [ (k2 + H 2 ) " Yc' 2 N(x)] u - 0. If, as we assume here, N(x) is a raonotonically increasing function of the height x, then there is an infinite set of eigenfunctions u (x) which van- ish at x ■ 0, tend rapidly to zero at infinity, x ■ oo, and satisfy the equation (25) u^ ♦ [k 2 - Yc' 2 N(x)J u - 0, where k are the eigenvalues of the operator -d /dx + yc N(x) and are therefore independent of k. We have then (26) r 2 - k 2 - k 2 . Since p positive corresponds to attenuated waves while \ negative corresponds to propagated waves, k is the cut-off wave number of the n-th mode. Below that wave number the wave is attenuated: 2 2 1/2 (27) -P-a- (k 2 -k 2 ) , and above cut-off it is propagated so that (28) -ir - P - (k-k 2 ) . The group velocity is found from (17) and (28), and is (29) v g - c /l-(k n A) 2 . We see that the situation is pretty much the same as in the case of parallel- plate waveguide except for the actual values of the cut-off wave numbers k 1 ,kp...etc. - 11 - Suppose that N(x) is identically zero between the ground and some small height. Then that strip the solution of the wave equation can be written down explicitly. From Eq. (25) we have, for sufficiently small x, (30) ik x -ik x n n u n - e so that the field is ■ u exp 1 'n n r i(k' " k n ) y J (3D exp ik x ♦ i(k 2 - k 2 ) y| n n — exp 2 2 1/2 " -ik n x + i(k -k£) y Thus we obtain near the ground the decomposition of the mode field into plane waves incident on and reflected from the ground and propagated in directions making an angle (Zf with the ground, where (32) - sin -1 (k n A) • Again as for parallel-plate waveguides, we obtain the following relation between the angle of arrival and group velocity for any mode: (33) v /c » cos Of. g This relation is independent of the charge density distribution N(x) and of the mode number. Since this model of the ionosphere is unrealistic we shall not calc:l»««. the cut-off wave numbers for any specific electron distributions. The problem is essentially the same as finding the energy states of an anharmonic oscillator with potential N(x). In most cases the phase integral method will give fair results - 12 - even for the lowest mode. Our main interest in analyzing this case, however, is in the general properties of propagation which will carry over to some ex- tent to the case of lossy ionopshere. Thus, in the latter case, we should ex- pect a finite number of modes with relatively small attenuation and the rest highly attenuated; we may expect the group velocity and the angle of arrival to be related in a way analogous to (33). U. Lossless ionosphere. Vertical polarization The main difference between the two polarization components in the lossless case is not in the boundary condition at the ground, but in the differ- ential equation. For vertical polarization we have » ♦ &*. - * 1/2 u- 1/2 ) ♦ r 2 > « - o. OU a • - - - -» The unusual simplicity of the case of horizontal polarization lay in our ability to split the coefficient of u, (k e+ T ) into two terms; one of these, k + P^, was independent of x, while the other, k (e-l), was independent of k. Now, how- ever, due to the extra term in (3U), we cannot do this. Furthermore the extra term is singular at the zero of e. This singularity did not appear in the case of horizontal polarization, where the only singularity on the real axis was at infinity. The presence of a singularity in the differential equation poses pro- blems which will not be discussed in this report. Apart from those difficulties we would still find that because the coefficient of u cannot be decomposed into a function of x and a function of k we lose the simple dependence of the propaga- tion constant Pon the wave number. - 13 - 5« Lossy ionosphere, Horiaontal polarization As was pointed out above, at low frequencies the complex dielectric constant is given very closely by Eq. (5). Hence the differential equation we have to deal with is (35) u^ + k 2 + P 2 + ikY N(x)/cv(x) u - 0. Again, as in the case of propagation of vertically polarized waves in a lossless ionosphere, we cannot expect the simple frequency dependence of the propagation constant given by Eq. (26) since the x-dependent terra also depends on k. Since N(x)/v(x) is a monotonically increasing function of x without bound we have a well-set eigenvalue problem, and the spectrum is discrete. If any doubts arise as to mathematical rigor, they can be overcome in most cases by writing instead of i in Eq. (35) exp i(^- + 6) and allowing 5 to tend to zero in the final result. In our calculations this limiting process yields the same result as taking 6-0 to begin with. Let us now introduce the exponential model of the ionosphere. We assume that (36) N(x) = N q e a , (37) -x/h v(x) » v e , o so that (38) where N(x)/v(x) - (N/v ) e 3 ^ , o' o -1 -1 " 1 (h/ ♦ h^) - Hi- Substituting thi3 in the differential equation (35), we obtain (39) n n + [k 2 ♦ T 2 ♦ ikY(» /%> e " X/h ] » " <>• In order to simplify the notation somewhat we shall introduce the following parameters: (UO) G 2 = k 2 + P 2 , (ul) 6 - kY(N Q /cv o ) h 2 . The latter parameter, 6, is small if the effect of the ionosphere at the ground is to be small. Eq. (39) can be solved exactly in terms of Bessel functions. By means of the substitution (U2) 4 - e X / 2h the differential equation is transformed into Bessel 's equation [11] (h3) u^ + I U 4 + J**S-+Ui6 u « 0. Thus the solution is the appropriate cylinder function of order (UU) m - 2iGh and argument (US) K - (±6) 1/2 e x/2h . The choice of the suitable cylinder function is dictated by the required behavior at infinity. Since the field must tend to zero, the Hankel function H (£) must be chosen. Expressing the solution in terms of Bessel functions of order +m and -m, we have -15- (U6) u - -i sinmn H (1) (C) • a" 1 *" JjK) - J (S). In order to find m, and therefore G, we have to impose the boundary condition at the ground. At the ground, however, the argument of the Bessel functions is small, so that one term of the power series expansion is sufficient to re- present the Bessel function accurately; then the wave function u becomes (U7) u ^ e iGx -Re- iGx , where (U) B - (-i6)- 2i0h -QiiE^ . P(l-2iGh) we may observe that if the dielectric constant had been (h9) e - 1 - yN/cov, the above analysis would still hold, but instead of -i6 we would have 6 in Eq. (U8). In order to satisfy the boundary condition at x ■ 0, the reflection coef- ficient must be unity. The equation (50) R - 1 has an infinite set of solutions. To order these solutions and assign to each some integer designation, we proceed as follows. First find the locus of all complex values of Gh for which the reflection coefficient has magnitude 1« It is obvious that the origin, Gh - is on the curve. Proceeding on the curve away from the origin, the phase of the reflection coefficient increases monotonically. Points corresponding to integral multiples of 2n satisfy Eq. (50). Let us use then, the ratio between the phase and 2n as the designation for a particular eigenvalue. - 16 - Having located one point on the curve, namely the solution of (50) corresponding to n « 0, let us consider this point further. G ■ implies that the wave is propagated parallel to the ground with phase velocity c» Furthermore, the incident and reflected waves, both traveling in the same direction, cancel each other at the ground. Hence they must cancel each other everywhere. That this is true near the ground is obvious from Eq. (U7). Hence G = is not really an eigenvalue and we should start counting with n = 1. Before proceeding with the investigation of the locus of |r| - 1 in the G-plane, let us see what this locus would look like if the dielectric constant were real, say as in Eq. (U9). We can readily see that in that case the locus is the real axis. The absolute value of the ratio of the gamma functions in Eq. (U8) is unity when G is real, since these two functions are complex con- jugates. The remaining factor would be a real number raised to an imaginary power, again having magnitude one. We would still have to choose between the positive and negative real axis. This choice, however, is purely arbitrary 2 for the purpose of determination of the propagation constant, since only G appears in the equation. Let us choose the positive real axis. The next step is to investigate the locus of |r| ■ 1 in the complex G plane. Due to the complexity of Eq. (U8) we have to look for suitable approxima- tions. Then, for sane values of parameter 6 we may check numerically the accuracy of our approximations. Since we are interested primarily in the first few eigen- values, the most interesting part of the curve is in the neighborhood of the origin. Let us introduce the dimensionless parameter (51) X - 3^ - iXg - 2Gh. - 17 - In terms of X, the equation of the curve is (52) (a «p[i §f -iUj-iXg) ra+x^+i^) ■ or (53) -Xg log 6 - - X^ + Re log r(i-x 2 - ix i ) Eq. (53) is the exact equation of the curve on which the absolute value of R is 1. In order to find an approximation for the curve in the neighborhood of the origin, we assume X- and X to be small and expand the last terra of (53) in a power series. Since (5U) a Tu+x) X » » -c, C - 0.577..., Eq. (53) reduces to (55) X g log 6 + - ^ - 2CX 2 - 0, or (56) _2 h 2C+log6 Thus we see that as a result of loss, the locus of |r| » 1 near the origin is pushed into the fourth quadrant. It starts as a radial line making a small angle with the real axis. Next we can investigate the behavior of the locus far away from the origin. - 18 - (A 0> 3 — * O i > O c _ ~ io * X o» ro U> UJ — • — ii o CO w l_ 3 o >*- o o - 19 - For large X we can use Stirling's formula for the gamma function of large argu- ment. When we do that we find that X, and X ? satisfy the following relation: (57) h -n/2 X 1 log6-21og|X|+2 ' Since X_ is obviously much greater than X , the derivation from the real axis is not very great. Having thus obtained approximations to the curve at both its ends, we should see how good these approximations are and what happens in the r egion when neither approximation is valid. For this purpose we investigated numeric- ally the case log 6 » -9 or 6 ■ 1.3x10 . This value of 6 is Jiigher than those we would encounter in most cases, so that the agreement in general should be better than in this case. The results are shown on Fig. 3« Since the curve does not deviate too much from the real axis, the modes corresponding to values of G to the left of the abscissa k will be propagated with little attenuation, while those to the right will be highly attenuated. It remains now to find on the locus to |r| ■ 1 those points which corres- pond to R ■ 1, i.e., the points at which the phase shift is an integral multiple of 2n. For this purpose we calculate the phase, i.e., the imaginary part of log R, where R is given by Eq. (U8). Here again the exact expression would have to be calculated numerically, but, since we are interested in the first few modes only, we shall make use of the power series expansion (58) $ ~ -£,_ log 6 ♦ £ X 2 -203^. This is the expression for phase $ in the X plane, near the origin. We need to know it on the locus |r| ■ 1, and so we make use of the relation (56), obtaining the phase in terms of Lj - 20 - i~-V l0 * 6 * 2C)+ ! X l 2C^6 (59) where - Xjte + n 2 Aa), a - -log 6 -2C For the n-th mode, (60) f - 2nn, so that (61) X^ 2nn (a+n 2 /Ua)' 1 and (62) X 2 * nn 2 (a 2 + h 2 /!*)" 1 . These two relations together with (51) and (UO) give the propagation constant . The numerical evaluation of the attenuation and phase constants is quite straight-forward now. We wish to obtain some information, however, about the frequency dependence of the attenuation factor and about group velocity. In order to avoid unnecessary complications we shall restrict ourselves to that range of frequencies in which the particular mode in question is not heavily attenuated. In view of the fact that 6 is very small we may neglect n /h with 2 respect to a . Thus we have (63) o n ~ g (1 .i £), so that <6U) *l * (g>* (i-i a > , - 21 - and (65) h a The range of frequencies is that in which k is greater than (nn/ha), and is not too close to it, say (66) k > 2nn/ha . i— i O The imaginary part of under these circumstances is much smaller than the real part. Thus we have (67) and n nn, 1 - (— ) v kha ; 1/2 (68) a n^ n2 n 2 /2h 2 a 3 p. We should remember that in the above relations a is a function of frequency. From (59) and (Ul) we have (69) a -leg 6-2C - log (cv o /YNh 2 )-2C - log k. If a is large, as we have assumed, the effect of frequency on a is very small and can be neglected for the sake of simplicity. We see then that in the pro- pagation region the exponential ionosphere has the same effect on phase and group velocity as a perfectly conducting plane sheet placed at the effective height w: (70) w ■ ah. The group velocity is obtained by differentiating (67) with r espect to k, neglecting variation of a - 22 - v* 'I 1 ' «=> ]* • For k sufficiently large the attenuation coefficient should vary approximately as k , or proportional to wavelength. Near the earth's surface the field can again be decomposed into two plane waves. This is a consequence of Eq. (H7)« Since G and | are not real, however, the plane waves are not homogeneous. But the attenuation is small and we may treat the plane waves as if they were homogeneous. The angle of arrival is then given by the same formula as in the lossless ionosphere, Eq. (32), and the same relation (33) holds between group velocity and angle of arrival. 6. Lossy ionosphere. Vertical polarization In the case of vertical polarization, in addition to the different bound- ary conditions at the ground, we have a modified differential equation. If the dielectric constant is the same function of height as in the preceding section, namely (72) e - 1 ♦ i(YN /cov Q ) e x/h , then the e which appears in the differential equation differs from e by (73) e*-e - -k 2 ^ (e^ •k 2 3 e x 1 e xx r t*2 ~r E In order to simplify the treatment we shift the origin to x • x , defined by x /h (YN /cov o ) e ° - 1 (7U) Z - X - x^ . o - 23 - In terms of z we have then (75) e - 1 + i e z/h , (76) e*-e - k" 2 h" 2 3 < e 2/h , 2 + i e Z ^ 5 W* } 2 TZF* 2 2 * The locus of k h (e -e) is shown on Fig. U. We see then that so long as kh is greater than unity, the difference between e and e is small compared to e. In fact is is also small compared to (e-l). So far we have been assuming that frequency is sufficiently low, at least in the calculation of eigenvalues. If we wish to neglect the term (76) in the differential equation, we must have kh > 1 and therefore sufficiently high frequency. Forturnately the approximation to the dielectric constant we have been using is valid below ca. 200 kc, while kh ■ 1 corresponds to ca. 2$ kc. In that range the low-frequency thoery used in this report should apply to the case of vertical polarization, with the effect of polarization on the differential equation neglected. We do, however, have to apply the appropriate boundary con- dition, i.e., u ■ at x ■ 0, so that instead of R * 1 we must have R ■ -1. This changes essentially nothing in our discussion of the last section, except that (77) $ - 2(n- |) n, and all the formulas of the preceding section follow, with (n- r-) substituted for n. 7« Applications We have shown here that horizontally polarized low-frequency radio waves are propagated between the earth and the ionosphere very much as in a parallel- - 2k - plate waveguide. The first few modes are propagated as if the ionosphere, instead of being a medium of gradually varying properties, were a perfectly conducting sheet located at the effective height w above the earth. This effective height w is simply related to the ionospheric parametersj the scale height of the varia- tion of conductivity and the values of the electron density and the collision frequency at some height. Since this model is lossless it cannot, of course, yield the attenuation coefficient. The latter can, if desired, be calculated from Eq. (65). Let us now calculate the effective height of the ionosphere. Let us take reasonable values of electron density and collision frequency at the height of 60 km: N(60) - 10 8 electrons/m 3 , v(60) - 10 sec." , and further let us take the respective scale heights h, and h 2 : h" U km, h 2 ■ 8 km, so that h - (h^ 1 + h^ 1 ) - 2 | km. With these assumptions we have 6 »k e a - - log 6 -2C ^19.3 -log k. In the low-frequency range k is of the order of l/e, and hence 6 is indeed small and a does not vary very much with even a substantial change of k. - 25 - Finally, the effective height w is, in the neighborhood of 25kc (a - 20), w - ah ■ 53.3 km. This height would not differ by more than a few percent over the whole very- low and low-frequency band. At I8kc we would expect the ratio v /c to be 0.976 in the second mode g (n - 3/2 in Eq„ (71)). This is in fair agreement with the experimentally ob- Tl2l served value 0.97 • The fact that the lowest mode was not observed cannot be explained on the basis of this thoery. In any case it should be clear that we deal here with low-order modes. Incidentally, the angle of arrival corres- ponding to the observed relative delay of about 3 /o is 1U , or considerably larger than that obtained by multiple-hop calculations. Perhaps it is possible to verify this figure experimentally. It is possible that the waveguide theory of low-frequency propagation provides an explanation of 'tweeks'L ■** *- -■ . Tweeks are very-low-frequency noise signals associated with lightning discharges. Following the discharge the frequency of the tweek decreases rapidly at first and then approaches an asymptotic value. This is just the behavior expected on the basis of the theory presented in this paper. The asymptotic value would correspond to the cut-off frequency of the mode. Furthermore, there is experimental evidence'- -"' L J that a 'tweek' is accompanied by its harmonics. But since the cut-off frequen- cies of the first few modes are integral multiples of the cut-off frequency of the lowest mode, the observed harmonics may really have been just higher frequen- cies arriving in a higher-order mode. The frequency-time relation for a typical 'tweek' was plotted by Burton and Boardman'- ^ f their curve is compared in Fig. U with computed values obtained from the parallel-plate waveguide model (using Eq. (71)) • We obtained the distance o o E > (0 3 ^ i o> ,_ _> c XT 0> 5 Q. ; s- ■ 5 E O 0) "£ ■o son with E V o O O .* O mpari eeks" 3 U (D <* K O u o ■i ii o 5 O ■o o-*- • «•- «*- o in r ro o r m in CM o CM - in io 0)i ui. Aousnbajj 1 1 1 r «■ CM O CD CD *• CM O rO rO ro CM CM CM CM CM CO CD - 27 - from the lightning discharge to the receiver by fitting the theoretical curve to the experimental one at 2.2 kc and assuming the cut-off frequency to be 1.6 kc. From this value of cut-off frequency the effective height is computed to be hi km; this is in excellent agreement with the effective height calculated from our theory, 53 *3 km. Indeed, the general agreement between the experimental and theortical curves is quite good. Acknowledgment The author wishes to express his gratitude to Professor B. Friedman for many helpful discussions. [l] Watson, G.N. [2] Schumann, W.O. [3] Schumann, W.O. DO Budden, K.G. H Budden, K.G. [6] Rydbeck, O.E.H, W Alpert, Ya. L. [8] [10] M Cl2] [13] CM Stanley, J.?. Stanley, J. P. Bremmer, H. - 28 - References - The transmission of electric waves round the earthy Proc. Royal Soc.,A 9£, 5U6 (1919), - Uber die Oberfelder bei der Ausbreitung langer, elektrischer Well en im System Erde-Luft-Ionosphare und 2 Anwendungen (horizontaler und senkrechter Dipol)j Z. fiir ang. Physik, 6, 35-U3 (195U). - Uber die Strahlung langer Well en des horizontales Dipols in dem Iufthohlraum zwischen Erde und Ionosphare, I| ibid, 6, 225-9 (195U); II, ibid ., 6, 267-271 (195U). - The propagation of a radio atmospheric, I •Phil. Mag., U2, 1, (195D; II, ibid .. ]j2, 1179 (1952). - The propagation of very low frequency radio waves to great distances} Phil. Mag, UU, 50U (1953). - On the propagation of radio waves; Trans. Chalmers Univ., Gothenburg, Sweden, No. 3b (19UU). - On the theoretical calculation of the field of low fre- quency electromagnetic waves over the earth's surface; Doklady Ak. Nauk 97, 629-32 (195U). - Ionospheric reflection of long radio waves j Can. J. Res., A 28, 5U9 (1950). - The absorption of long and very-long waves in the iono- sphere} J. Atm. Terr. Phys., 1, 65-72 (1951). - Terrestrial radio waves; Elsevier Publ. Co., Amsterdam, 19b 9 p. 133. Magnus, W. and Oberhettinger, F. - Special functions of mathematical physics; Chelsea, New York, 19U8. Brown, J.N. Potter, R.K. - Round-the-world signals at very low frequency; J. Geophys. Res.,5U, 367 (195U). - Analysis of audio-frequency atmospherics; IRE Proc, 39 , 1067 (195D. Burton, E.T. and Boardman, E.M, - Audio frequency atmospherics; Bell System Tech. Jour., 12, U98 (1933).