Radio Science

Transmission of waves near the lower oblique resonance using dipoles in the ionosphere

Authors


Abstract

[1] The observations in the OEDIPUS-C experiment of bistatic propagation along the lower oblique resonance cone between a separated transmitter and receiver in the ionosphere are explained through detailed calculations of the radiated field of the V-dipole antenna used in the experiment and by a novel theory of the receiving antenna under resonance conditions. Unexpectedly high values of 25 kHz signal observed at the resonance and its structure agree well with calculations of the transmission between the exciter and receiver, when antenna layout and dispersive properties of the plasma at resonance are taken into account.

1. Introduction

[2] The rocket experiment OEDIPUS-C (OC) produced observations of the transmission and reception of waves between separated payloads along a propagation path close to the lower oblique resonance cone. 25-kHz waves were received at a distance of 1200 m from a transmitter [James, 2000] at a height of several hundred kilometers in the auroral ionosphere. The existing theory of whistler-mode radiation from a dipole antenna was tested in a first quantitative analysis of these measurements. The whistler mode is the right-hand polarized cold-plasma mode occurring in regions 7 and 8 of the CMA diagram [Stix, 1992]. Electric fields near the group resonance cone were significantly stronger than in other directions, as expected for quasielectrostatic propagation near the resonance cone. Modulation of the received signal level with the spin phase of both the emitting and receiving dipoles was consistent with the expectation that the predominant electric field of waves propagating near the resonance cone is linearly polarized and at right angles to the group resonance cone. Signal levels induced on the receiving dipole, however, were about a hundred times greater than theoretical predictions based on some conventional procedures for the electromagnetics of transmitting and receiving antennas.

[3] The present paper reports successful attempts to reconcile the aforementioned disparity between experiment and theory. The salient features of the OC experiment are reviewed in section 2, as well as the relevant observations. The organization of the theoretical calculation of the transmitted levels falls logically into two parts. In Section 3.1, we develop the formulas for the electric field radiated by the OC “Double-V” transmitting dipole under the quasistatic approximation using the methods described by Mareev and Chugunov [1987]. Then, in section 3.2, we present the relevant forms of the expressions developed recently by Chugunov [2001] for the open-circuit voltage induced on a receiving dipole located close to the cold-plasma resonance cone surface. The data-theory comparison following, in section 4, shows that the disparity reported by James [2000] disappears when the essentials of dipole reception are correctly treated.

[4] Figure 1 is the whistler-mode refractive-index surface for 25 kHz under the relevant ionospheric conditions. The resonance surface in wave-vector (k) space is at an angle of about 85° with respect to the ambient terrestrial magnetic field B. Therefore the group resonance cone is at an angle γres = 5° away from B. The received signal voltage V is computed for dipole positions just inside the cold group cone, as shown in the inset cartoon, and just outside. The curve labeled “n/1000” effectively changes the x and y scales to show the refractive index curve for n values of about 1000. Analysis has shown that a range of wave number k values corresponding to n between about 100 and 1000 make the dominant contribution to V. Electromagnetic k values, such as those near the point of inflection indicated by the angle θs, need not be included in the evaluation.

Figure 1.

The refractive index surface n for the parameters of the experiment, shown in Table 1. At the point of inflection on the n curve, the ray has the maximum angle θs with B, 14°. Waves of interest have much larger n values like the ones shown on the n/1000 curve. There, the transmitter-receiver separation lies very close to the group resonance cone, which makes an angle γres = 5° with B. The curve labeled “n/1000” gives the value of n divided by 1000 and shows the refractive index curve for values of n up to 4000. This curve is broken at values where collisionless damping is not negligible. In the inset drawing, the transmitter-receiver total separation is about 1200 m. Their separation perpendicular to the magnetic field B is about 100 m.

[5] The theoretical development in section 3.2 draws attention to unique subtleties of resonance cones. The cone makes it possible for an antenna, even under CW excitation, to radiate a wide spectrum of waves in k space near the k-resonance surface. The correct description of a receiving dipole, reacting to this rich incident k spectrum, requires some care. In applying the Chugunov [2001] formalism to present OC circumstances, we arrive at an expression for the open-circuit received voltage V = E Leff, the product of the radiated electric field E in the absence of the receiving dipole and the dipole effective length Leff. The expressions for and evaluation of the latter show that it can attain values of up to 100 times the physical half-length L of receiving dipole. Thus, the behavior of the receiving dipole in the presence of the resonance-cone k spectrum is shown to be critical to the understanding of the aforementioned disparity.

[6] Although the detailed treatment of resonance-cone wave reception has been lacking up to now, the finding of enhanced Leff values is not surprising. The reciprocity principle, used by Chugunov [2001], has long been interpreted to imply a correspondence between radiation-pattern peaks of emitting antennas and Leff enhancements of receiving antennas. Since the enhancement of signal levels near the resonance cone has been known for some time [James, 2000], we should not be surprised to find large values of Leff, as we have in OC. Notice that even for electromagnetic whistler-mode propagation, Leff enhancements can be observed [Sonwalker and Inan, 1986].

[7] The OC results and the ensuing analysis refocus attention on the interpretation of antenna measurement resulting from propagation known or suspected to be resonance-cone. The interpretation of the amplitudes of spontaneously created resonance-cone waves may be critical to the assessment of plasma instabilities that create them.

2. OC Experiment Description and 25-kHz Results

2.1. Experiment Details

[8] The OC sounding rocket [James, 2000] was launched at the Poker Flat Research Range in central Alaska on November 7, 1995. Starting at about 1.8 minutes after apogee, experiments on plane-wave propagation were carried out between the two free-flying halves (subpayloads) of the double payload. The upper subpayload contained a digitally controlled radio transmitter called the High Frequency Exciter (HEX) and the aft subpayload contained a corresponding Receiver for Exciter (REX) that was tuned to the same frequency. Both subpayloads had four tubular beryllium-copper monopoles forming a crossed dipole configuration. This geometry was required for spin stability of both subpayloads. The tip-to-tip length of each of the crossed-dipoles 2L was 19 m on HEX and 13 m on REX. The two terminals of the “V-dipole” antennas were the inner endpoints of two orthogonal monopoles. That is, pairs of adjacent orthogonal monopoles were connected together at the subpayload body to produce the “V” shape.

[9] The observations of bistatic propagation at 25 kHz in the whistler mode occurred at times after launch (TAL) between 700 and 760 s, during which separation went from 1181 to 1196 m. Parameters of the experiment shown in Table 1 were obtained from various payload measurements. Swept-frequency HEX-REX recordings provided ionograms from which the ambient plasma frequency fpe was scaled. The Plasma Probe on the lower subpayload provided the electron temperature Te and a confirmation of fpe (D. D. Wallis, private communication). The electron gyrofrequency fce was calculated using the IGRF1995 coefficients. The various angles defining the subpayload orientations and their separation vector also were derived from onboard magnetometer and video camera systems.

Table 1. Experiment Parameters
ParameterValue
Wave frequency, f0.025 MHz
Plasma frequency, fpe0.300 MHz
Gyrofrequency, fce1.200 MHz
Temperature, Te5000 K
Debye length,λD14.4 cm
Driving point current, I05 mA
Monopole length, L9.5 and 6.5 m
Separation distance, r1190. m
Separation angle, θ05.0°
Payload altitude650 km

2.2. RF Voltages Observed by REX

[10] Figure 2 contains a series of spectra of the signal voltage, for most of the down-leg portion of the flight when dipole-to-dipole propagation was studied. Each spectrum is produced by a 600-μs rectangular pulse of 25-kHz oscillation. The spectra are of the RF voltage V at the REX-input antenna terminals. The results for the frequency bandwidth of interest, 0-50 kHz, are plotted. The blackness of the pixels is proportional to the spectral density of V2, as indicated by the inset graded scale. As discussed by James [2000], the separations and widths of the spectral main and side lobes are within 20% of the values expected for an ideal rectangular 600-μs pulse.

Figure 2.

The history of the frequency spectrum of the 600-μs square pulses with a carrier frequency of 25 kHz. The overall evolution of the spectrum is dictated by the requirement that the group resonance cone lie along the transmitter-receiver separation direction.

[11] During the subinterval 700 to 750 s, a clear HEX signal is seen near 25 kHz. The diffuse trace then rises in frequency over the next 100 s, going past the 50-kHz limit of the frequency scale at about 850 s. Waves at all of the signal frequencies in this event, from around 25 kHz through 50 kHz, are known to be excited by the HEX because of the transmitted spectrum observed at small transmitter-receiver separations earlier in the flight. James [2000] defined a resonance-cone frequency which meets the requirement that the subpayload separation direction lie along the group resonance cone. The effect of the resonance cone in this transmission experiment is to enhance signals with frequencies near the resonance-cone frequency. Because parameters change with time, the resonance-cone frequency changes with time. James also explained the one-on/one-off nature of the signals in 700 < TAL <750 s as caused by the combined spin effects of the transmitting and receiving dipoles.

[12] The receiving dipole voltage V observed as a function of time is plotted in solid line in Figure 3. Dipole spin effects again are obvious, causing modulation of roughly 200 mV between successive maxima and minima. This diagram reproduces the signal levels for the frequency bin in Figure 2 centered at 25 kHz. Previously, a theoretical link calculation was carried out using the quasistatic theory of Kuehl [1974] for the radiating OC dipole and an assumed Leff of the receiving dipole equal to its real length [James, 2000]. When an RF current magnitude (5.4 mA) for the transmitting dipole determined from equivalent-circuit analysis was applied to such a theory, the calculated V values were much lower than observed. These calculations are shown with the dot-dash line close to the TAL axis in Figure 3. When the driving point current was used as a fitting parameter in Figure 3, the best-fit curve was found only with an unrealistically large value of current, 300 mA.

Figure 3.

Observed (solid line) and theoretical (broken and dot-dash lines) open-circuit voltage V on the receiving antenna versus time at 25 kHz. The broken-line curve was obtained with formulas (13) for V and (6) for the radiated field. The dot-dash line was obtained from a theory [James, 2000] that assumed that the effective length of the receiving dipole equals its real length.

[13] The rest of this paper describes how we have reworked the link calculation in order to understand the foregoing disparity. We call upon the quasistatic theory of the radiating dipole of Mareev and Chugunov [1987] accounting for radiation from all four monopoles of the double-V dipole connected to the HEX. On the receiving end, the results of Chugunov [2001] are applied to derive Leff of the receiving dipole for the case when it is located close to the resonance cone. This approach is able to account for the strong signals observed at 25 kHz.

[14] This analysis is restricted to signals right at the pulse carrier frequency of 25 kHz. It is clear from the data used here and from other OC data that they contain information about the distribution of wave energy in the neighborhood of the resonance cone. See the discussion of “Pulse Spectra” and of Figure 6b in the work of James [2000, pp. 1345–1346]. The generalization of the receiving dipole theory to all positions of that antenna with respect to the resonance cone and to arbitrary pulse spectra constitutes a considerable elaboration of the theory, and so is not undertaken in this paper.

3. Theory

3.1. Radiated Field

[15] It is well known that the electric field in the vicinity of a resonance cone is derivable from an electrostatic potential to a high level of accuracy. Because the cold-plasma point-source field diverges on the cone surface, thermal and electromagnetic corrections, which are of the same order of magnitude, must be included in the dispersion relation to make the field finite [Mareev and Chugunov, 1987; Fiala and Karpman, 1988]. In what follows, we shall use the expression given by Mareev and Chugunov for the only important field component, namely the component perpendicular to the cone surface. These conditions allow us to employ a shortened method for the field calculation, which uses an asymptotic expansion of the dispersion equation.

[16] The structure of the field near the resonance cone is easy to find by converting to a frame of reference with a τ axis directed along the resonance surface and a ξ axis perpendicular to it, i.e. by turning the coordinate system xyz through the angle γres around the y axis, orthogonal to the plane containing B and the radius-vector r. Our coordinate system has its origin at the center of the transmitting dipole (ξ = 0), and ξ is positive inside the cone. In this system, the ξ - component of the electric field is much greater than τ - component. Spatial harmonics of the field are separated, giving the main contribution to the electromagnetic field in the coordinates (τ, ξ). The phase velocity of quasi-potential waves is directed along the electric-field direction while the group velocity is directed along the resonance cone (which is just the reason for the resonance cone formation). In the quasi-electrostatic approximation, the electric field for a given current can be easily found by the Fourier transform of the Poisson equation (see Mareev and Chugunov [1987] for details). After a simple algebraic procedure and integration over kτ (using a pole in the dispersion equation) and ky (using a saddle point method which selects harmonics with the group velocity directed from the radiation point to the point of observation) the electric field strength is written in the form of the integral over the normalized wavenumber x = kξL. Note that in the frequency band under consideration, a continuum of harmonics with kξ values from 0 to ∞ gives the contribution to the resulting field at the observation point:

equation image

The coefficients depending on frequencies are given here for the lower oblique resonance in a strongly magnetized plasma, for equation image where ω = 2πf, ωLH is the lower hybrid resonance angular frequency, ωpe is the angular electron plasma frequency and ωce is the angular electron gyrofrequency. The characteristic electromagnetic wavelength λ is much greater than the antenna half-length L, that is,

equation image

where c is the speed of light. (2) ensures resonance excitation of quasi-potential waves in the electromagnetic wave spectrum. The normalized charge spectral density (in k space) due to a current Io injected into the V-dipole with a triangular current distribution (uniform charge density in real space) is

equation image

and the argument of the exponential function in (1) is

equation image

Here and henceforth, ξ and τ are replaced by their normalized values ξ = ξ/L and τ = τ/L. γres is the resonance cone apex half angle (Figure 1), and φ denotes the azimuthal angle that the V-dipole axis of symmetry makes with the plane defined by external magnetic field and the observation point, i.e. the τ − ξ plane. The V-dipole lies in the plane perpendicular to the magnetic field. The second term in the phase ϕ(x) is an electromagnetic correction, and the third is a thermal correction with R as an “effective” Debye radius. The fourth term is proportional to s, the electron-neutral collision frequency divided by the excitation frequency. This term was found to be negligible in our case and so was dropped.

[17] Notice that the field can be expressed analytically if the electromagnetic and collisional corrections are dropped. Moreover this approach is entirely sufficient to describe the field between the resonance cone and B at ξ > 1, and gives a complete picture of the wave field. In this respect, we are interested in the region of parameters and coordinates τ and ξ, where the electromagnetic correction can be neglected. We introduce a new variable equation image and write the electric field as follows:

equation image

For ξ ≥ 1 we can use an asymptotic approximation obtained by the method of stationary phase for the integral as the parameter equation image. We obtain

equation image

where

equation image

and

equation image

The expression (6) shows that the radiation is a wave packet of plasma waves with the characteristic wavelength

equation image

which depends on space coordinates ξ and τ . rD = νTepe is the Debye radius. The equation image function reflects the modulation of the wave packet due to the interference of waves coming from the four arms of the V-dipole antenna.

[18] The amplitude of the radiated electric field E as a function of the distance ξ is shown for φ = π/6 in Figure 4 as a representative example. The result of the numerical computation of the complete integral expression (1) is plotted in solid line. The broken-line curve gives the asymptotic values from (6). The numerical evaluation of (1) is found to be in very good agreement with the analytical evaluation (6) for ξ ≥ 1.

Figure 4.

Amplitude of the electric field E excited by the V-dipole against distance ξ from the cold plasma resonance cone, normalized by antenna arm length L = 9.5 m, for azimuthal angle φ = π/6. The numerical calculation from (1), in solid line, and the asymptotic expression from (6), in broken line, are in good agreement for ξ ≥ 1.

[19] It is convenient to rewrite (6) in ‘local’ spherical coordinates, τ = r, ξ = rθ, with angle θ measured from the resonance cone surface. Now the expression for the field takes the form

equation image

Here, r is the distance between the transmitting and receiving points divided by L.

[20] Some features of the radiation field should be pointed out. First, the field is composed of propagating waves for positive values of the angle θ only, whereas for negative values of this angle, the field is evanescent. Therefore, the resonance surface θ = 0 is the boundary of radiation and “shadow” regions. Further, the radiation field amplitude is proportional to θ−(1/2), i.e., it grows steeply as the resonance cone is approached. That is, the radiation pattern of a dipole antenna changes substantially under resonance conditions. Finally, the dependence upon the angle φ reflects the V-shaped geometry of the antenna.

[21] We checked numerically that, indeed, the electrostatic approximation is good at the OC separation distance of 1190 m, i.e. the long wavelength part of the k spectrum corresponding to whistler-mode waves does not contribute significantly to the E field. We also checked that the Eξ component is much greater than φ component of the electrostatic field, the τ component being still lower.

3.2. Receiving Antenna Under Resonance Conditions

[22] As is well known, for a short dipole in vacuum, equation image, where λ0 is the free space wavelength, the r.m.s. voltage V induced on the dipole terminals by the electric field E of an electromagnetic wave is

equation image

where F(Θ, Ψ) is the radiation pattern as related to electric field in the θ, φ direction (non-normalized value). In this experiment, the receiving monopole length L = 6.5 m. In a complex medium like a magnetoplasma under resonance conditions, calculation of the voltage induced on the receiving antenna is a complicated electrodynamic problem, even for a short dipole. As seen in the preceding section, the generated quasi-potential wave packet has a complicated structure. In particular, the radiation pattern, the phase- and group-velocity relations are distinctly different from the vacuum case.

[23] The expression corresponding to (10) for a magnetoplasma under resonance conditions is required in the framework of a full transmitter-plasma-receiver link calculation. A relatively simple way of calculating the voltage induced on the receiving antenna is based on the use of the reciprocity theorem. This procedure is well known for random fields. For a magnetoplasma under resonance conditions the power spectrum of the noise voltage equation image induced at the frequency ω is given by Andronov and Chugunov [1975]. This calculation uses the reciprocity theorem and the fluctuation-dissipation theorem, and it leads to the following expression which is the integral analogue of the well-known Nyquist formula, in a case of a homogeneous magnetoplasma under resonance conditions:

equation image

where

equation image

is the longitudinal dielectric permittivity with an electromagnetic correction equation image is the Fourier-component of the distribution of a unit amplitude current at the receiving antenna; δ is delta-function; equation image is the spectral density of electromagnetic energy of the incident radiation in the wave-number space; εαβ is the dielectric tensor; and Ξ = fpe/f.

[24] Formula (11) is valid in general for arbitrary nonequilibrium radiation incident on the receiving antenna. Therefore, it can be applied to the regular (deterministic) radiation as a special case of nonequilibrium radiation. We require the r.m.s. voltage value, equation image where Δω is the receiver frequency bandwidth.

[25] Multiplying (11) by Δω and integrating over the wave numbers, we obtain the following expression for the voltage induced on the receiving antenna [Chugunov, 2001]:

equation image

Here equation image is the angle between the direction to the receiving antenna gap and the resonance cone, and equation image denotes the azimuthal angle that the receiving V-dipole axis makes with the plane defined by external magnetic field and the observation point. The receiving V-dipole lies also in a plane perpendicular to the magnetic field. Equation (13) has been derived for the plasma wave mode from the corresponding equation for the voltage induced obtained from equations (12) and (14) of Chugunov [2001].

[26] The equation (13) states that the voltage induced for open circuit is the product of the incident wave field, antenna length and the excitation coefficient, which is the electric field of the radiated wave normalized to the field of the point dipole. The excitation coefficient of the plasma wave mode is seen from the formula (9) while the factor equation image appears due to normalization to the field of the point dipole. The two terms inside the absolute value sign in (13) depend on the orientation of the two linear dipoles composing the receiving V dipole. The sum of these two terms provides the addition of the voltages corresponding to the projection of Eξ along each linear dipole. Putting the expression for the electric field of a radiating dipole into (9) one can see that the r.m.s. voltage induced on the receiving antenna is proportional to the electric field of the incident wave, the characteristic size of the V-shape dipole, and the radiation pattern as related to the electric field near the resonance cone. Note that the excitation coefficient in (13) corresponds to the excitation of a plasma wave by the transmitting V-dipole, as this part of the electromagnetic field is being excited and reradiated by the receiving antenna.

[27] Since V = E Leff, the effective length is given by (13) divided by E. We use (9) and (13) as the basis of comparison with the experimental data.

4. Comparison of Measurement and Theory

[28] The geometry of the radiation and receiving points are shown in Figure 5. The specific geometry and wave parameters of the OC HEX-REX link have particular consequences for the theory developed here. With reference to Figure 5, a characteristic magnitude of the resonance cone spreading due to the finite size of a radiating antenna is Δξ ≈ L ≈ 10 m. Consequently, the receiving antenna, separated from the radiating antenna by r ≈ 1.2 · 103 m, subtends a small angle Δθ1 = Δξ/r ≈ 10−2. Furthermore, the resonance cone angle, or the angular extent of the radiation region at f = 25 kHz is γ ≈ f / fpe ≈ 10−1. Therefore, the linear size of the region where the receiving antenna is able to record the wave field is given by Δξreceiverr · γ ≈ 102m, so that equation image. These estimates allow us to conclude that the frequency 25 kHz is revealed clearly in the transmitter-receiver experiment (Figure 2) for the following reason. At this frequency, the actual geometry promotes large signal magnitudes because the receiving antenna is located in the radiation region sufficiently close to the resonance cone. At other frequencies, the receiving antenna either turns out to be too far from the resonance cone, or lies, at least partially, in the “shadow” region. In either case, a relatively small signal is expected.

Figure 5.

Geometry of the layout of transmitting and receiving antennas in the experiment. The x-z plane contains the locations of the centers of both the transmitting and receiving V-dipoles. Two radiating monopoles are positioned along the x axis. The other two are perpendicular to the page at the origin. Two receiving monopoles are shown with heavy lines labeled “receiving point.” The three parallel lines sloping upwards to the right define the intersections with the x-z plane of three transmitter resonance cones, one at each of the extremities of the x-aligned monopoles and the third at the drive point, or origin.

[29] In the formula (13), E(r, θ = Δθ0) is the amplitude of the radiated field at the receiving point, as given by (9). The angle γ in Figure 5 gives the direction between radiating and receiving points. The angle Δθ0 = γres − γ, where γres is the angle of the resonance cone. In addition, the minimum value of the ξ coordinate is ξmin = min[−Lcosφ, −Lsinφ], giving the position of the cones coming from the ends of the transmitting antenna, keeping in mind that ξ is measured from the resonance cone centered on the drive-point of the V-dipole. From this, it is evident (compare Figure 5) that the received signal reaches a maximum when the receiving antenna is in the radiation zone next to the resonance cone. This corresponds to the angle Δθ0 being Δθmax ≈ 2L/r. This is the situation that we assume in the interpretation of the experimental data. As the antenna goes farther inside the radiation region, the signal gets lower as both the amplitude of the field decreases and the angle Δθ0 grows. As the antenna goes into the “shadow” region, its aperture becomes smaller, and the signal is exponentially small.

[30] To compare these theoretical results with the observations, we plot the open circuit voltage at the antenna terminals as given by (13) against time in the Figure 3 with a broken line. As argued above, the receiving antenna is positioned just next to the resonance cone (see Figure 5) and inside it, to receive the maximum signal at 25 kHz as revealed by observations (compare Figure 2). Accordingly, we take Δθ0 ≈ 2L/r (approx. 0.01), i.e. ξminL, at a distance r ≈ 1.2 km from the transmitter. The azimuthal angle equation image changes with time in relation to spin periods of both the payloads (11.4 s and 11.9 s respectively for HEX and REX) and the orientation of the payloads is measured in the experiment. The open-circuit voltage reflects the multiplication of radiation patterns of both the transmitting and the receiving antennas. In our case of V-dipoles, each pattern shows five resonance cones emanating from the four antenna tips and the gap. This means that the voltage changes rapidly between maxima of about 300 mV and minima of a few mV on time scales of about 1 s. In the experiment the voltage is measured twice in 3 s intervals and the measurements are 0.5 s apart. In order to compare the measured and computed voltages, we evaluate (13) at the time of measurements and we use a best fit procedure to match calculated values (broken line) with measurements (solid line). The result is remarkably good agreement, both in amplitude and spin phase, taking into account that we are looking at one minute of results of a space measurement.

[31] The observed received voltage in Figure 3 diminishes abruptly at about 755 s. This is simply because this plot shows the voltage in a 521-Hz Fourier-analysis bin centered at 25 kHz. At 755 s, local fpe jumps upward, forcing the resonance cone frequency to a higher bin, and leaving little spectral power in the 25-kHz bin [James, 2000].

5. Conclusions

[32] The results of the OEDIPUS-C two-point experiment on propagation along the lower oblique resonance cone in the ionosphere are well explained by the theory that takes into account both the antenna layout and dispersive properties of this plasma mode. The high voltage observed on the receiver side stimulated a novel approach to the study of receiving antennas under resonance conditions. We have confirmed what was inferred from the OC experiment, namely that the effective length of an antenna can exceed substantially its geometrical length. In the particular case observed at 25 kHz, Leff ≈ 30L. Moreover, the good agreement between observations and theory implies interesting possibilities in wave diagnostics of the ionospheric plasma, where the measurement of parameters requires careful consideration of the whole transmitter-plasma- receiver link.

Acknowledgments

[33] This work was supported by NATO PST.CLG 974764 grant, Czech Ministry of Education ME 356, INTAS grant N 00-465, and the Russian Fund for Basic Research project 00-02-17758. The authors also thank graduate students Alexey Soldatkin and Václav Jásenský for performing numerical calculations.

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