On the amplitude and phase velocity of VLF and LF e.m. waves in the spherical earth-ionosphere waveguide

Journal of Atmospheric andTerrestrial Physics, 1972, Vol.34,pp.877-392. Pergamon Press.Printed in Northern Ireland

On the amplitude and phase velocity of VLF and LF e.m. waves in the spherical Earth-ionosphere waveguide JA. L. AL’PERT, D. S. FLIGEL’, Academy

0. V. KAPUSTINA

and I. N. ZABAVINA

of Sciences of the U.S.S.R., Izmiran, Moscow, and The Leningrad University, U.S.S.R.

(Received 19 July 1971; in revised form 20 October 1971) Ab&r&--This paper contains the results of theoretical investigations of the electrical field radial component l&l and ph ase y as well as the mean phase velocity 8 and the differential velocity wdof VLF and LF waves in the spherical Earth-ionosphere waveguide. It is taken that the ionosphere is homogeneous and isotropic and the conductivity of the lower boundary-the Earth is infinity. The calculations are made for frequencies f = 6 -50 kHz and for distances D = 100-10000 km. Two theoretical methods are used: the hop-method for D = 100-5000 km and the mode-method for D = lOOO-10,000 km. It has been shown that for distances D = lOO4000km the hop-method is more convenient and for distances D .w 2000-10,000 km the modemethod is more useful; for the overlapping distances the both methods give practically identical results. The influence of different number of hops on the field amplitude and the phase has been considered. The detailed analysis of the dependence of phase velocity I) on distance has shown that the maximum deviation of 0 from C is of the order of 3 per cent at distances D M lOO1000 km and 0.2 per cent when D w 10,000 km. A detailed considering of the dependence of e/c on frequency for different fixed distances has led to new phenomena unknown till now. At different fixed distances D for some ‘character frequencies’, namely, for the selected ionospheric parameters at frequencies f = 12.9-13 kH z, D m 515 km; f = 22-22,05 kHz, D FJ2700 km; ,‘,:26.,26.9 kHz, D w 525 km; f = 42.5-43 kHz, D rw2900 km, the phase velocity e/c changes

1. THE AMPLITUDE AND PEASE OF THE FIELD IN FLIGEL et al. (1971) the results of calculations

of the amplitude, phase and phase velocity of VLF and LF waves were considered in a spherical waveguide with homogeneous isotropic boundaries, obtained by means of the mode-method (GYUNNINEN and RYBACHEK, 1966 ; RYBACREK, 1968) on frequencies f = 5-50 kHz at distances from the source D M lOOO-10,000 km and for different conductivities of the Earth (a,) and of the ionosphere (bJ. For distances D < 1000-3000 km the calculations by the mode-method are too complicated: a great number of modes should be summed for receiving sufficiently precise results. Therefore for these distances it is more convenient to use the so called expansion on ‘rays’ or ‘hops’. In this expansion every member, describing the field at a definite distance, may be interpreted as a wave, which is reflected several times from the Earth and the ionosphere (GYUNNINEN and ZARAVINA, 1966 ; ZABAVINA, 1969). In this case the radial component of the field is E

z

=

27hV &&o~ v _ m sin 0

3001/w Rod0

where W is the emitter power in kW, k, is the wave number in the free space, R, is the Earth radius, 0 is the angular distance between the source and the point of a77

878

JA. L. AL’PERT, D. S. FLIQEL’, 0. V. KAPUSTINAand I. N. ZABAVINA

observation

and V = V, +

2 ,V, W&=1

(2)

is the attenuation function. In equation (2) ‘CT,, is the field of the wave, which propagates from the emitter to the receiver along the Earth’s surface without any reflecdescribe m-multiple hops, i.e. rays tions. The other members of the series (2)-V, reflected from the ionosphere. The members of the series with numbers m = 1, 2, . . . and so on we shall call further as the first, the second hops and so on. If the source and receiver are on the Earth surface then (3) and V,=2exp

exp (iG?)Al”(t)B”-l(t)

s4 (*3W2P

[w,‘(t)

-

dt

aL7W,(t)12 ’

where

A(t) =

Wl’W - 4) + QiWl(t - do) T,'V

qq

=

-

W,,,

are

+

w,'(t)

-

ew2w

w,'(t)

-

a,~dt)

Xi,* are the sines of incident surface,

4J

the

Airy

qiw2p

-

d,)

'

(4) '

angles of the rays on the ionosphere functions,

W$,

their

and the Earth’s

derivatives,

k, = kOd&

k, = Ico< and ci, E, are the permittivities of the ionosphere and the Earth. The results of calculations of the modulus and argument of V(D) for inhomogeneous on radius ionosphere were given earlier in ZABAVINA (1969), however without detailed study of the phase v(D,f). In this paper we give results of calculations of the radial component of the electrical field E,, of the additional phase y which determines the deviation of the phase from its value in free space:

and of the differential and mean phase velocities in the range of frequencies f = (5-50) kHz at distances D w (lOO-10,000) km from the source. At distances

On the amplitude and phase velocity of VLF and LB e.m. waves

LI 2000

11000

6000

_-

111 SOG

879

0 D,km

Fig. 1. The field strength IE,(D)I mV/ m vs. the distance for the frequencies f = 5, 20, 30 and 50 kHz. Solid lines-results of calculationsby the hop-method; dotted lines-by the mode-method.

500-5000 km these calculations have been compared with the data obtained by the mode-method (FLIGEL'etal.,1971). It is also considered the influence of the number of hops on the accuracy of calculations of the amplitude and phase of the field on fixed frequencies. As in FLIGEL' etal. (1971), a model of the Earth-ionosphere waveguide with the homogeneous isotropic ionosphere and the homogeneous Earth has been considered. The values of the ionosphere conductivity cri = 2.5 x lo3 CGSE and the Earth conductivity og = co have been taken, the height of the waveguide h = 70 km. Some results of j&(D)/ are given for f = 5, 20, 30 and 50 kHz, obtained by the

JA. L. AL’PERT, D. S. FLIOEL’, 0. V. KAPUSTINAand I. N. ZABAV~NA $rad 4 2 0 -2 -9

0 -2 -4 -6

-a

Fig. 2. The additional

phase v vs. the distance for the frequenciesf = 20, 30

and 60 kHz. The notations are the s&meas in the Fig. 1.

mode-method for D = 500-10,000 km (dotted lines) and by the hop-method for D = 100-5000 km (solid lines) in Fig. 1.* It was shown that the lower the frequency the broader the distances, where the results of the both methods coincide. So on f = 5 and 15 kHz the amplitude values coincide within the all overlapping interval of distances ; on f = 20 and 30 kHz they coincide when D 2 1500 km and on f=50kH z (as 1‘t is seen from Fig. 1) it occurs when D 2 2000 km. The analogous picture is obtained for the additional phase y(D) (see Fig. 2). From Fig. 1 it is also seen that in the nearest zone at distances D M 100-1000 km very deep interferential maxima and minima of field amplitude are observed. With the distance increase these oscillations are smoothed. With the frequency growth the interferential picture intensties. The analysis of all the results of amplitude and phase calculations has shown, * The calcul&ions of I&(D)\ were fuElled for W = 1 kW.

On the amplitude and phase velocity of VLF and LF e.m. waves

881

Fig. 3. The influence of different numbers of hops on [E,(D)] and phase y(D) for f = 30 kHz.

that on frequencies 5 < f G 30 kHz the mode-method is advisable to use starting from D 2 1000 km and on frequencies f > 30 kHz from D 2 2000 km. The hopmethod may be used for the whole range of frequencies 5-50 kHz, starting from the nearest distances D = 50-100 km up to D M 3000-4000 km. At greater distances for receiving sufficient accuracy it is necessary to add a great number of members of the series (2) (for f = 5 kHz, m > 10-11). The number, n of modes, which is necessary to take into consideration, decreases with the increasing distances, at D M 4000-5000 km, n N 2-3. For B N 1000-4000 km, where both methods are suitable, for the higher frequencies it is more convenient to use the hop-method and on lower frequencies, the mode-method. In Fig. 3 on f = 30 kHz it is shown the influence of different number of hops on the modulus of the attenuation function 1VI and the additional phase t. The

882

JA. L. AL’PERT, D. S. FLIGEL’, 0. V. KA~USTINAand I. N. ZABAVINA

notations

x on every curve show the number of members considered in series (2). 0,1,2 For example 2 means that in calculations of this curve the ground wave V, 0x63

and three members m = 1, 2, 3 have been taken into account in series (2). It is seen from Fig. 3 that on f = 30 kHz for the considered interval of distances it is quite possible to take into account the ground wave and five hops. The member m = 6 practically does not change the values of 1VI and y. With the account of the sixth hop the amplitude changes in the third decimal place after the point and the phase, approximately on one degree. If only three hops are used, then, for instance, on D = 5000 km the error in determination of 1VI is of the order of 15 per cent, and for the phase, about 5 degrees. The analogous consideration of the influence of different numbers of hops which was made on other frequencies has shown that on f = 10 kHz and at D m 100-1000 km it is sufficiently to use (2-3) hops, for D M 3000 km, 6 hops, and for D = 4000-5000 km, 8-9 hops (the next hop changes I VI less than on some O-1 per cent, and the phase y-on some parts of a degree). Onf = 50 kHz for the D w 100-1000 km m N (l-3) ; for D M 3000 km m N 5 and for D w 4000-5000 km, 7 hops are to be considered. 2. THE PHASE VELOCITIES The additional phase y has been used for determining wdphase velocities : fl=

C

l+Y-Yoc D-Dow

and

the mean d and differential

vd = 1 + -

C

,

--AY c ADw

where y. are the values of y close to the source, more precisely when D, -+ 0 ; w is the circular frequency w = 271-f,c is the light velocity in free space, Ay is the phase difference in two close points and AD is the corresponding difference of distances. There are some difficulties in calculations of the phase velocity: it is necessary to know the dependence of the absolute value of y on the distance D. Therefore first of all it is necessary to do detailed calculations of the phase through such intervals of D in order not to miss the phase change on 2~. Second, the distance D must be sufficiently small because otherwise it is impossible to determine exactly the ‘initial phase’, yo. The second condition was impossible to fulfil in the paper of FLIGEL’ et al. (1971), where the mode-method was used and calculations were carried on starting only from distances D w 500-1000 km due to insufficient numbers of modes for closer distances. In the present paper the calculations of the phase have been performed starting from D m 100 km, where according to our calculations y,, M 0’. It differs from zero not more than on 2-5 degrees. Depending on the character of phase variations on different frequencies the intervals of AD = 1, 2, 5, 10, 25, 50 and 100 km have been chosen, which allow to watch precisely the continuous variation of the phase. Finally we have obtained more detailed dependences of the phase y (D, f = Const)

On the amplitude and phase velocity of VLF and LB e.m. waves

883

than earlier by the mode-method in FLIGEL’ et al. (1971). These results show a very interesting peculiarity of the field phase behaviour in the spherical EarthIt turned out that there are some rather narrow intervals ionosphere waveguide. of frequency Af = (fi-fi), w h ere starting from some critical distance D m D,,. the functions y(D) change their character when the frequency changes from fi to fa. So on frequency f N fi when D > D,, the values of y(D) increase with the distance growth, and on frequency f ‘ufi the phase y(D) decreases with distance increase. When the interval Af decreases the curves goes away more steeply and the phase difference Ivl(fi2D) - y2(f2,01 ~PProximates to the value of 2rr. The variation of the amplitudes with distance on these two frequencies I.ZZl(fi, D)l and IE2(f2, D)l practically coincide and have a deep minimum when D M D,,, where a sharp change of the character of phase curves occurs. This effect is illustrated in Figs. 4-7, where the curves of [E,(D)], y(D) and a(D)/ c are plotted for f = 12.9 and 13 kHz (Fig. 4), f = 22 and 22.05 kHz (Fig. 5), f = 26.7 kHz (Fig. 6) and f = 42.5 and 43 kHz (Fig. 7). From Fig. 4 it is seen, that at D < D,, w 515 km the phases y(D) of two waves which have very close frequencies 12.9 and 13 kHz coincide. When D M 515-517 km on fi = 12.9 kHz yl(fi, D) sharply increases and on fi = 13 kHz y2(f2, D) sharply decreases. The less fi distinguishes from f2 the more narrow is the interval of distances where this transition occurs. It is illustrated in Fig. 8(a) by comparison of curves on frequencies f = 12, 12.5, 12.8, 12.9 and 13 kHz. The analogous picture is observed on f = 22 and 22.05 kHz for D,, m 2600-2700 km (Fig. 5). On f = 26.7 and 26.9 kHz this effect takes place for DC7 w 525-527 km (Fig. 6) and on f = 42.5 and 43 kHz-for D,, M 2800-2900 km (Fig. 7). The similar picture probably will occur on f M 48-50 kHz, but we have not analysed this frequency range in detail. The analogous effect takes place for model of the waveguide, chosen in ZABAVINA (1969), where the inhomogeneous on height ionosphere is considered starting from h = 64 km, and the lower boundary-the Earth represents a two-layered conductive surface. In this case the rapid transition of y(D) occurs on other values off and D,,, namely, on frequencies f = 11.5-11.6 kHz at D,, es 388-392 km (Fig. 8(b)). This sharp change of the phase can be explained simply as a result of summing of several harmonic oscillations which can be represented as vectors (see Fig. 8(c)). It is seen that at f = fi = 12.9 kHz a resultant vector has a left-hand rotation when the distance D increases from 510 to 520 km. At the frequency f = fi = 13 kHz the resultant vector has a right-hand rotation in the same range of distances. As a result the phases on these two frequencies differs on 277. Naturally, the effect of the rapid change of the additional phase y(D) when D m D,, results in analogous behaviour of the mean phase velocity e(D). The curves of B(D)/c for the mentioned above ‘characteristic’ intervals of frequencies 12.9-13 kHz, 22-22.05 kHz, 26.7-26-g kHz and 42.5-43 kHz have shown that the greatest differences in velocities for two close frequencies, i.e. the differences SC/c = I(fi/c), - (C/C)~~~are: 4.5 x 1O-2 for the first interval of frequencies, 4.2 x 10-a f& the second, 2 x 1O-2 for the third and 2.4 x 10e3 for the fourth frequency interval. In Fig. 9 are plotted the curves fi(D)/c of the mean phase velocity for f = 5, 30,40 and 50 kHz, at which the above mentioned effect is not observed. The curves

884

JA. L. AL’PERT, D. S. FLIQEL’, 0. V. KAPUSTINA and I. N. ZABAVINA

On the amplitude end phase velocity of VLF and LF e.m. waves

885

886

JA. L.

AL’PERT, D.

S. FLIGEL’, 0.

V.

KAPUSTINA and I. N.

ZABAVINA

sad 4 0 -I -2 -3 -4 -5

f,=t2*9nHz

fi13~Hz

D =5201cm

D=520km

Fig. 8. The additional phase y(D) vs. the distance for different frequencies (a) for homogeneous boundaries of the waveguide a( = 2.5 x lo3 CGSE, Gg = w, (h = 70 km); and (b) for inhomogeneous boundaries of waveguide. (c) the rotation of a resuItant vector E, vs. the distance for f = 12.9 and 13 kHz.

On the amplitude and phase velocity of VLF and LF e.m. waves

887

% 1.016

4001

4001

100

100

MO

09’

0

2000

4000

6000

sooo

ILam

Fig. 9. Theratio0(D)/cv~.thedistanceforf = 15, 30, 40snd50 kHz. Solidlinesresults of calculations by the hop-method. Dotted lines-by the mode-method.

of the differential phase velocities wJD)/c for f = 15, 30 and 40 kHz are given in Fig. 10. As in Figs. 1 and 2 the solid lines are the results received by the hop-method and the dotted lines by the mode-method. It is seen, that at D w 100-200 km 5 is up to 1000 km, e/c naturally close c; when the distance increases approximately quickly increases, reaches the maximum at D w 800-1000 km and then slowly decreases with the distance increase. The analysis of the obtained theoretical results has shown that the maximum deviations lAf~/cj = 16/c - 11 on the frequency range f = 10-50 kH z are of the order of 2 per cent for D = 100-1000 km about O-4 per

888

JA. L. AL'PERT,

D. S. FLIUEL’, 0. V. KUUSTINA md I.N. ZABAV~NA

099 0

2000

4000

6000

8ooo I.I,Kttl

Fig. 10. The ratio v,#)/c of the differentialphase velocity vdto the velocity in the free space c vs. the distance forf = 15,30 and 40 kHz. The notation is the same &s in Fig. 9.

cent for D = 3000 km and they are not more than 0.1-0.2 per cent for D M 10,000 km. The differential phase velocity changes irregularly with distance and frequency. To receive these changes in more detsils it is necesssry to calculate v,JD,f) through very small intervals of AD. In Fig. 10 the curves are calculated with the interval of AD = 100 km. Maximum differential phase velocity deviation from c, namely lAGI = IQ/C - II w 8 per cent for D = 1000 km. When D w 10,000 km lAv,/cl N IA@/ w 0.3 p er cent. The comparison of values vd, calculated for smaller intervals of AD = 20 km with those given in Fig. 10 is plotted in Fig. 11 for f = 10 and 40 kHz (the dotted lines are the results for AD = 100 km, and the solid lines

On the amplitude and phase velocity of VLF

and LF e.m. waves

889

Q/C I

II

I

IllI

I

II

I

II

1.02

400

099

096

0.911

099

096

0

Fig. 11. The ratio wd(D)/c vs. the distance for f = 10 and 40 kHz. Solid linesresults of calculations with distance intervals AD = 20 km. Dotted lines-with AD = IOOkm.

for AD = 20 km), It is seen that the curves ‘v& obtained for AD = 100 km lose a number of fine details. The results given above show that the character of the functions ycf) or Bdf) at fixed distances D depends on the fact whether D is greater or smitller than minimum value of distance DET. In the first case the curves ydf) or fi/ccf) change continuously, in the second case they represent stepped broken curves; the number of steps and character of the discontinuity depend on D, for which these curves are calculated. The corresponding curves of C(f)/c for two fixed distances: D = 500 km (i.e. D < D,es,min) and D = 3000 km (D > Dcs,min ) are given in Fig. 12. At D = 500 km the curve C/C oscillates with frequency, however, the discontinuity of sharp-jump

890

JA. L. AL’PERT, D. S. FLIGEL’, 0. V. KAPUSTINA and I. N. ZABAVINA

1.020

i.015 4,040 i-005 IO00 0.995 0990

(b)

Fig. 12. The ratio of mean phase velocity B to c vs. frequency for distances D = 500 and 3000 km.

type is not observed on this curve. When D = 3000 km e(f)/c, c has five characteristic parts, where C(f)/c has jumps .* Maximum change of C/C on f = 12.9-13 kHz at D = 3000 km, namely, (~Y/c),=~~~n~- (fi/~),=~~~n~ m O-008; on other frequencies where jumps occur, these differences are smaller. By the way, we should notice, that if the phase is artificially shifted on 27m radisns in the places of discontinuity, the parts of the curve of Q/C are also shifted, forming a continuous dependence of 5/c on frequency. In conclusion it should be noticed that the jump-effect of the phase and mean phase velocity with frequency in the Earth-ionosphere wave-guide had not been described till now in the literature. It may be connected with the fact that in published works (see for example JOHLER and BERRY, 1964) there were not made such detailed phase calculations depending on distance and frequency as in ours, which naturally does not allow to discover this effect. As to the experimental data on measuring phase velocity, which are published in the literature (see for example AL’PERT and FLIGEL’, 1970), usually the observations by airplsnes or at fixed points were made on one or two fixed frequencies. Therefore all these data collected together (as it was done, for instance, in FLIGEL’ et al. (1971) and AL’PERT and FLICEL’ (1970)), obtained at different times and conditions can not expose this effect. For experimental discovering of this effect it is necessary to do some special and detailed measurements on a rather great number of frequencies. The question of methodics of those experiments and their practical value is not discussed here. * The last frequency interval f = 45-50 kHz was not analysed indetails (see the Appendix).

On the amplitude and phase velocity of VLF and LF em. waves

891

Acknowledgenzent-The authors wish to thank G. I. MAKAROVfor useful discussions,and J. A. BARINOVAand L. V. LISINAfor their help in preparingthis paper and calculations. REFERENCES AL’PERTJA. L. and FLIOEL’D. S.

1970

FLIGEL’D. S., KAPUSTINA0. V. and RYBACEKEK S. T. G~UNNINENE. M. and RYBACHEKS. T.

1971

1966

GYUNNINENE. M. and ZABAVINAI. N.

1966

JOF~LER J. R. and BERRYL. A. RYBACHEKS. T.

1964 1968

ZABAVINAI. N.

1969

Propagation of ELP and VLP Waves near the Earth. Consultants Bureau, New York & London. Qeomag. & Aeronomk 11, 102. Diffraction and Radio Wave Propagation Problems, No. 6, p. 115. Issue of the Leningrad State University (LGU). Difraction and Radio Wave Propagation Problem, No. 5, p. 5. Issue of the Leningrad State University (LGU). NBS Monograph 78 (October 1964). Diffraction and Radio Wave Propagation Problem, No. 7, p. 152. Issue of the Leningrad State University (LGU). Diffraction and Radio Wave Propagation Problems, No. 5, p. 5. Issue of the Leningrad State University

(LGU).

APPENDIX The calculations

fulfilled by us last time showed that in the frequency range f = 40-50 kHz, which was not calculated in detail before (see the footnote on p. 889) there are not one but two discontinuities of the phase and phase velocity of sharp-jump type. As a result the dependence @J(f)/c at D = 3000 km will have six characteristic parts where @df)/c has jumps (in Fig. 12. C(f)/c has only five jumps). So the reader must consider Fig. 13(d) instead of 12(b). Besides the Fig. 13 (a, b, c) define the dependencies of IE,(D)I, v(D) and @(Q/c given in Fig. 7.

10

892

JA. L. AL’PERT, D. S. FLIUEL’, 0. V. KAPUSTINA and I. N. ZABAVINA

LLd 0 -2

-4 -6 -8 -iO -iZ

% me +008

0 G

f 004 1402 ’

IMO

3ow

2mo

D,kn. %

im

mo I.005

iW 0995 IO

20

30

40

50

f, km. Fig. 13. (a), (b) and (c) Correction to Fig. 7. The functions jE,(D)I, y(D) and ~(D)/c vs. the distance forf = 42.5 and 43 Wz. (d). Correction to Fig. 12(b): The ratio of mean phase velocity 0 to c vs. frequency for distance D = 3000 km.