Propagation of a radio wave pulse in cold plasma

2. Propagation of a Pulse in an Isotropic Plasma

[3] Propagation of the pulse is described by a wave equation

(1)

where E is the electric field strength, c is the speed of light in the medium, z is the direction of propagation, t is the time, and P is the polarization of a unit volume of the medium. In accordance with the model of the medium with free charges [Pamyatnykh and Turov, 2000; Vinogradova et al., 1979], P is described by

(2)

In equation (2), e and m are the charge and mass of an electron, respectively; N is the electron concentration, and n is the effective collision frequency which takes into account energy losses by electrons due to collisions with neutral molecules and ions.

[4] At the boundary of the half-space z 0 where the pulse propagates, the field E is written as

(3)

Here, w = 2p f; f is the carrier frequency; and A(0;t) is the pulse envelope at z=0.

[5] The leading edge of the pulse always propagates with the velocity of light in the medium. Accordingly, the field E will be sought in the form (k=w/c=2p/l is the wave number)

(4)

In equations (1) and (2), variables are changed as

(5)

Taking into account equation (4), we get

(6)

(7)

Let us compare the magnitudes of the first and third terms in the left-hand side of equation (6) for the pulse with a carrier considered here. Its characteristic duration t_p satisfies the inequality

(8)

and the pulse itself occupies on the z' axis the interval

(9)

Therefore the estimate

(10)

is valid, and the first term in the left-hand side of equation (6) can be neglected. Note that inequalities (8) and (9) are also fulfilled for wideband pulses [Varakin, 1970] and their special case, i.e., pulses with a linear frequency modulation. Therefore estimate (10) is valid for these pulses as well.

[6] Since the position and velocity of an electron cannot change instantaneously, the conditions

(11)

are fulfilled in cold plasma at the moment of arrival of the pulse at point z' . Then the solution of equation (7) is given by

(12)

Let us substitute solution (12) into equation (6). If we take into account estimate (10), the equation for the pulse envelope A(z';t) acquires the form

(13)

By applying the Laplace transformation with respect to variable t' to equation (13), we get

(14)

The designations we use here are

is the Laplace image of the initial envelope; d = d(z')=w₀²z'/2c; w₀²=4p e² N/m; w₀ is the plasma frequency; and it is assumed that A(z';0)=0. It is evident that the latter condition is fulfilled for the pulse whose initial envelope is zero at t=0

[7] By performing inverse Laplace transformation, we obtain

(15)

J_k(x) is the Bessel function here and in what follows.

[8] The solution of equation (15) is written for the initial envelopes which will be used below as examples to illustrate the character of distortions of radio wave pulses in a cold isotropic plasma. A biexponential pulse with the envelope in the form ( A₀; a; b are the numbers)

(16)

is a good approximation for pulses with different steepness of the leading and trailing edges. By substituting equation (16) into equation (15) and changing the variable m= (q/t')^1/2 in the obtained expression, we get

(17)

where the a -dependent term is written as

(18)

and the term that depends on parameter b is obtained from equality (18) by replacing a by b.

[9] The initial envelope of the so-called sine wave pulse is given by

(19)

Let us present the envelope (19) in the form

(20)

where

and substitute equation (20) into equation (15). By performing the operations similar to those described above, we obtain the expressions for a deformable sine wave pulse

(21)

where

(22)

Let us now derive the solution of the problem for a rectangular pulse

(23)

whose distortions, according to equation (23), are described by

(24)

In order to obtain the f(z';t') function, it should be borne in mind that when a

0 and b

simultaneously, the bi-exponential pulse transforms into a step pulse of height A₀, the value of A remaining zero at t=0. By performing appropriate transformations in equations (17) and (18), at t'>0 we get the envelope of a step pulse in the form (A₀=1)

(25)

When t' increases, the problem on the incidence of a step pulse on a semi-infinite homogeneous medium transforms into the problem on propagation of a plane wave in a homogeneous medium. Let us demonstrate that solution (25) satisfies this requirement. Let us again turn to solution (15), which for a step pulse acquires the form

It is easy to see that relations (25) and (25a) are equivalent. Let t'

in equation (25a). In this case the integral in the left-hand side reduces to the tabulated one. Its calculation gives

Here, t=g(w)z' is the optical depth, and g(w)= w₀²n/w² c is the absorption coefficient of the plasma with respect to power at the carrier frequency (at n

w ). The first term in equation (25b) describes a decrease in the amplitude of the plane wave, and the second term describes an increase in the phase of the plane wave when it passes a distance z' [Ginzburg, 1967], which is what we set out to prove.

Figure 1

Figure 2

Figure 3

[10] Analysis of the literature on propagation of radio wave pulses in plasma has revealed, among other things, that a pulse with a rectangular envelope is most frequently considered as an initial pulse. For this reason, major attention will be given below to this type of pulses. Figures 1 and 2 differ only by the durations of the rectangular pulses; the parameters of the medium are taken to be approximately corresponding to the conditions in the ionosphere [Kirillov and Kopeikin, 2003]. In combination, they illustrate the character of distortions of a pulse caused by, in addition to t_p, the path length and the absorption effect. At t_p=10^-4 s (Figure 1), the shape of the envelope remains close to a rectangular one up to considerable distances from the source (~2000 km), and its distortions manifest themselves in oscillations in the field magnitude about an average level determined by the optical depth of the path. Elongation of the path is accompanied by an increase in the relative amplitude and characteristic period of the oscillations. As the pulse becomes shorter by an order of magnitude, all other conditions being equal (Figure 2), the character of distortions of the envelope appreciably varies and manifests itself now, first of all, in a pronounced deformation of the leading and trailing fronts. The fronts gradually spread, and beginning from some distance the pulse cannot be regarded as rectangular. From the point of view of the spectral representation, the change in the character of distortions of the envelope should be attributed to broadening of the spectrum of the emitted pulse, so that within its limits the frequency dependences of the refractive index and absorption coefficient of the medium become appreciably different from the linear ones. The effect of the rectangular pulse duration on its distortions can be understood in more detail from Figure 3, in which curves |A|² are compared for four magnitudes of t_p at the same temporal scale, all other conditions being equal. Figure 3 also shows that the velocity of propagation of the pulse in plasma is appreciably lower than the velocity of light and that the velocity of propagation of the leading edge of the rectangular pulse is not determined by the pulse duration. Let us derive an analytical expression for the pulse propagation velocity. To this end, we use solution (25). We compute the integral in equation (25) by parts successively an infinite number of times using the known relation for Bessel functions

(26)

As a result, we get

(27)

We assume that inequality 2(d t_p)^1/2

1 is fulfilled, which is typical of ionospheric paths (see the caption to Figure 1), and use the asymptotic representation of the Bessel function at large values of the argument [Korenev, 1971]

(28)

After substitution of equation (28) into equation (27) and summation of the series obtained in this way, the expression for A(z';t') acquires the form

(29)

It follows from equation (29) that on the z' axis the pulse is concentrated in the vicinity of the point to which the minimum of the modulus of the radicand in the denominator of the right-hand side corresponds. The coordinate of this point satisfies the condition

(30)

from which we infer that the leading edge of the rectangular pulse propagates with the velocity

(31)

According to equation (31), the effect of collisions can exert an influence on the propagation velocity if magnitudes of n are comparable to the carrier frequency, i.e., only under the conditions when the propagating pulse is strongly absorbed. For the parameters of the medium and pulse indicated in the captions to Figures 1, 2, and 3, the inequality w²

n² is fulfilled. In this case we have

(32)

Figure 4

Figure 5

It follows from equation (32) that at a constant path length, all other conditions being equal, the delay time of a pulse, as compared with a hypothetical case of its propagation with the velocity of light, increases with decreasing carrier frequency and increasing electron concentration. Along with this, solution (25) takes into account the change in the envelope at the end of the path caused by variations in the parameters f and N indicated (Figures 4 and 5).

Figure 6

Figure 7

[11] Figures 6 and 7 illustrate the character of distortions of a bi-exponential pulse (see equation (16)) and sine wave (see equation (19)) pulse for the parameters of the problem given in the caption to Figure 2. Comparison of the curves shown in Figures 6 and 7 with the curves presented in Figure 2 leads to the conclusion that in the case of propagation under similar conditions distortions of both the bi-exponential and sine wave pulses manifest themselves to a much lesser degree. From the point of view of the spectral representation, a reduction in their distortions is naturally interpreted as the consequence of narrowing of the pulse spectrum due to smoothing of its initial envelope.