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Keywords:

  • Doppler;
  • moving media;
  • special relativity;
  • wave propagation

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References

[1] The exact computation, the exact modeling of the propagation of electromagnetic signals through inhomogeneous moving media, is an old theoretical and measuring/data interpretation problem, because in a lot of practical problems, such as the accurate space-time determination in satellite positioning systems, the occultation measurements, or the accurate tracking of the interplanetary probes, the electromagnetic wave (signal) traverses moving, inhomogeneous media, where the moving velocity of the media and the media itself are inhomogeneous in space, i.e., the Earth's high and low atmosphere, the planetary atmospheres, the interplanetary matter in the inner and outer solar system, etc. In other important cases of electromagnetic wave propagation, as the propagation in inhomogeneous media or the propagation of short impulses (ultrawide-band (UWB) signals), the application of the method of inhomogeneous basic modes (MIBM) was successful. In this paper it is demonstrated that the application of MIBM in the case of inhomogeneous moving media produces an exact, full-wave solution of monochromatic or UWB signals propagating through these media. The solutions are valid inside the special relativity.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References

[2] The correct description, the correct modeling of the propagation of monochromatic, quasi-monochromatic or ultrawide-band (UWB, arbitrary shaped) signals in inhomogeneous and moving media has a basic importance in space experiments, in Earth investigations, in astronomical research, in wireless communications and in some other applications. To reach this goal, i.e., to the correct description of this propagation problem, it is necessary to have a real, full-wave solution of Maxwell's equations in moving, inhomogeneous media. However, in earlier approximations this full-wave solution was not presented.

[3] Measurements in communication investigations, in space research and in radio astronomical experiments [e.g., Adam, 1948; Birkemeier et al., 1968; Bird, 1976; Goldstein, 1969; Higgs, 1960; Jacobs and Watanabe, 1966; Sadeh et al., 1968; Woo et al., 1976] demonstrated the importance of a complete full-wave solution, and in this moment it is impossible to exclude that the effects caused by the signal propagation through the interplanetary, moving, inhomogeneous media in the outer solar system from the spacecrafts to the Earth has an important role in the “Pioneer anomaly” [see, e.g., Anderson et al., 1998]. To clarify the possible role of different factors in these measurements it is necessary to derive a correct, complete full-wave solution of Maxwell's equations in this important (special relativistic) case. An overview of the problems considering in these cases of wave propagation is given by Ferencz [1979]. The investigation of the frequency shift and line broadening with limited accuracy was applied earlier in a ray-tracing approximation for the quasi-monochromatic signals, the “relativistic ray tracing” [Ferencz, 1980a], some results on the interpretation of measured data are given by Ferencz [1980b]. Some results of the application of the classic scattering theory on the interpretation of data measured in terrestrial communication experiments are given by Birkemeier et al. [1968]. A method was developed for the investigation of the rotation of the signal polarization [Ferencz, 1978a, 1978b] but without the derivation of a complete full-wave solution. The determination of the complete, full-wave signal form propagating through these complex moving media when the signals are monochromatic or have arbitrary shape and the result must be valid inside the special relativity, i.e., the moving velocity of the media comparing to the observer is not very small, has not yet been solved.

[4] In some practical applications, such as in the detection of rotating rocket bodies in missile defense tasks or in stealth technology, the effects caused by moving and rotating targets are important. Therefore the investigation of the effects appearing on the signals propagating, reflecting or scattering in moving complex media is very actual. Some works have been made recently in this field [e.g., Li et al., 2010; Qiu et al., 2008; Zhang et al., 2009]. In these works the definite main goal was the determination of the Doppler frequency shift and line broadening in the case of real man-made targets and by using monochromatic (or quasi-monochromatic, i.e., radar type) source signals or excitation. Therefore in these very well based works the motion of the medium, the motion or spinning of the targets have low speed comparing to the velocity of the light (c), and no real UWB solution was needed. For example, in the work by Zhang et al. [2009] the description of the Doppler effect has a classic, nonrelativistic form and the spinning rate in the example is around 0.5 Hz, which is a high spinning rate in practice, but it is low speed comparing to the speed of light. In this case the goal is not the derivation of the propagating complete waveform, however, the determination of the Doppler bandwidth caused by the spinning target. The situation of Li et al. [2010] is similar, in which the spinning speed remains in the <2 rad/s domain. In the work by Qiu et al. [2008] it is important that the target (the medium) is layered and complex (anisotropic, bianisotropic), and in this work the derivation of the waveform of the propagating, monochromatic-type signal is solved. However, the motion, the rotation velocity is limited also, it remains inside the classic, nonrelativistic domain. (Another common special aspect of these works is the fact, that if somebody will be close to higher velocities in these cases, then it will be necessary to use a general relativistic description because the rotating systems are noninertia, accelerating systems.) Therefore the determination of the monochromatic (quasi-monochromatic) or UWB signals propagating through complex, inhomogeneous media, which have (comparing to the velocity of light) high and inhomogeneous moving velocities in space and which have complex (anisotropic, bianisotropic) character, remained unsolved.

[5] In the last years, however, with the exclusion of the motion of the medium traversed by the propagating signals, it was possible to derive precise full-wave solutions of monochromatic and UWB signals from the Maxwell's equations [Ferencz et al., 1996, 2001; Ferencz, 2005] with a lot of successful applications of these theoretical results mainly in space research [e.g., Ferencz et al., 2001, 2007; Lichtenberger, 2009; Ferencz et al., 2009]. The key step to derive these new solutions was the correct application of the method of inhomogeneous basic modes (MIBM) (see details given by Ferencz [1978c], Ferencz et al. [2001], Ferencz [2004], and Ferencz [2005]). A short summary of the MIBM is given in the Appendices A and B. The MIBM is based on the physical fact that it is sure that the final resultant signal, the complete signal, i.e., the sum of the “basic modes” (or any other sum or integral form of a solution) is a solution of the Maxwell's equations, however, these parts of the final resultant (e.g., an independent inhomogeneous basic mode, or the independent monochromatic components of a signal, which is written by Fourier series or Fourier integral) separately are not solutions and could not be solutions of the Maxwell's equations even if a medium is linear [Ferencz, 2004; Ferencz, 2005]. This was and is an important step in the derivation of new and correct propagation models because, citing Einstein's words: “Any serious consideration of a physical theory must take into account the distinction between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates” [Einstein et al., 1935].

[6] The physical concept of the MIBM was verified in the case of nonmoving, inhomogeneous media, of stratified media and of propagation of the arbitrary shaped, i.e., UWB, quasi-monochromatic and monochromatic signals. In the following we present an application of the MIBM in the case of propagation in inhomogeneous moving media in which the velocity field of the motion is varying in space too, however, it does not vary in time.

2. Initial Conditions and Formalism for Derivation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References

[7] We will use the well known and generally accepted formalism of the special relativity and the Lorentz transformation [see, e.g., Laue, 1955; Synge, 1965; Schmutzer, 1968]. Therefore let the K′ be the coordinate system comoving with the moving media at a given point. The media itself and the moving velocity distribution could be inhomogeneous in space respectively; however, they are constants in time in the comoving systems. Let us derive the electromagnetic signal at the receiver (sensor, observer) resting in the K laboratory coordinate system. The signal source (transmitter, etc.) could be comoving either with the K′ coordinate systems or with the K laboratory coordinate system. In the first case the first (initial) K′ system is defined as the comoving system with the signal source, independently from the moving of the media. The geometry is illustrated in Figure 1. Then we can use the Lorentz transformation as it is well known (see also Figure 2): The Lorentz transformation tensor is

  • equation image

if the axes of the K′ and K systems are parallel respectively, and the motion velocity of K′ in K is equation image = (v, 0, 0), i.e., the velocity equation image parallel with the x′ and x axes, a = v/c and j = equation image.

image

Figure 1. The general situation of propagation in moving inhomogeneous media.

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image

Figure 2. The common definition of the comoving (K′) and laboratory (K) coordinate systems. In the t = t′ = 0 moment the O and O′ coincide.

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[8] If the direction of equation image is arbitrary, i.e., equation image = (v1, v2, v3), then the v value in a = v/c is

  • equation image

and

  • equation image

The formulas of the vector and tensor transformations using the aik transformation tensor are well known also, as follows:

  • equation image

Here it is true that ariark = δik and airakr = δik, where the δik symbol is the Weierstrass δ - delta symbol, i.e., δik = 0, if ik, and δik = 1, if i = k.

[9] We will use the Maxwell's equations in the following form:

  • equation image

where equation image, equation image, equation image, and equation image are the electric field strength, the electric displacement, the magnetic induction and the magnetic field strength, respectively. equation image is the three dimensional current density, ρ is the electric charge density, c is the velocity of light in vacuum and equation image is the three dimensional differential operator. (It is valid in cgs system and in every comoving inertia coordinate system.)

[10] In the following we will use the well known four-dimensional formalism and in this case we shall use the four-dimensional electromagnetic tensor symbolism as it is known [see, e.g., Synge, 1965; Schmutzer, 1968]. If the electromagnetic tensors are

  • equation image

then the form of the Maxwell's equations is in general, special relativistic form including every effects of the motion

  • equation image

which is valid in every comoving or non comoving, inertia coordinate systems, and where σi is the four-dimensional current density

  • equation image

In (6) Ji are the components of the complete three-dimensional current density and ρ is the electric charge density. The Fik [LEFT RIGHT ARROW] Gik relation is defined by the constitutive relations in known manner in the coordinate systems comoving with the media, and in very complex form or it is not known, if the coordinate system is not comoving with the medium [see, e.g., Marx, 1952, 1953; Ferencz, 1979].

[11] It is well known that depending only on our decision it is possible to incorporate the effects of all charge motions, of all signal-medium interactions into the current density and charge, i.e., into the four-dimensional current density, and in this case FikGik. As it is known the signal-medium interaction contains the currents induced by the electromagnetic field in the medium, the circular currents induced by the electromagnetic field in the medium, the charge harmonic oscillations, i.e., the polarization effects caused by the electromagnetic field, etc., and the excitation signals, which generate the electromagnetic field. In this case all of these effects appear in the σi, which σi in this case naturally differs from the σi in (5). Then we can write the Maxwell's equations in a modified form, i.e.

  • equation image

We will use these equivalent formulas in the following derivations, first the equation system (7) and in the second case the equation system (5). We shall use these two derivations, because it is well known that the derivation of the medium-signal interactions in the K laboratory coordinate system is very problematic in the case of complex media [see, e.g., Marx, 1952, 1953; Kong, 1975]. However, the MIBM will produce a simple solution of this problem in which it is not necessary to derive the constitutive relations of the moving media in the K laboratory coordinate system.

3. Solution of the Maxwell's Equations by MIBM Using Equation (7)

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References

[12] The general structure of the propagation can be seen in Figure 1. We can write the Maxwell's equations in the coordinate systems K′ comoving with the media and in the laboratory coordinate system K as follows:

  • equation image
  • equation image

Let us write the Maxwell's equations valid in system K with the electromagnetic field tensor appearing in the comoving system K′. We know from (2) that

  • equation image
  • equation image

Applying (9) in system K, the form of the Maxwell's equations will be

  • equation image

[13] The most simple and simultaneously the control case is the special situation, in which the medium traversed by the signal is homogeneous (and time-invariant in the comoving system K′) and the velocity field of the medium is homogeneous also. Let us see first this case.

3.1. The Case When the Medium Is Homogeneous and the Moving Velocity of the Medium Is Homogeneous, i.e., Uniform

[14] In this case the medium traversed by the signal is homogeneous and time-invariant (in the comoving coordinate systems K′) and its velocity field is uniform and time-invariant, too. Therefore we have only one K′ system and the aik components of the Lorentz tensor are constants, independent of space and time. Therefore we can rewrite the equations (10)

  • equation image

Using in (11) the Weierstrass relation, i.e., ariark = δik, or ariari = 1, etc., and the known relation that

  • equation image

a new form of (11) will be

  • equation image

and knowing that the ari tensor is nondegenerative in every cases, then in the first equation of (11) we shall have nonzero terms for the electromagnetic field components only if the index t is t = s and the multipliers of ari are equal each other; the situation in the second equation is similar, where ariaskatl, is nondegenerative in every cases.

[15] Therefore finally the form of (11) is

  • equation image

as it is trivial, as it was known earlier in this homogeneous case and it is necessary because the Maxwell's equations are invariant from the sense of inertia systems.

[16] In the following for the definition of the inhomogeneous basic modes we shall use the equation forms (11)) and (13), based on the experiences presented, e.g., by Ferencz [2004, 2005].

3.2. The General (Inhomogeneous) Case Using Equation (7)

[17] In this general case beside the Frs(xs) and σr(xs), or Fik(xk) and σi(xk), the aik tensor coefficients also depend on xk or xs, i.e., aik(xk) or aik(xs). Let us start by using the equation system (10)

  • equation image

From this equation system

  • equation image

If we compare the equations (15) and (11), then a formal similarity appears between the first equation in (11) and the second and third terms in the first equation in (15); and between the second equation in (11)) and the second, fourth, and sixth terms in the second equation of (15). We can use these formal similarities in the definition of the inhomogeneous basic modes.

3.3. Definition of the Inhomogeneous Basic Modes

[18] Let us define the inhomogeneous basic modes as the solutions of those parts of (15), which parts do not vanish in the homogeneous case, presented in point 3.1, i.e., let the inhomogeneous basic modes be the solutions of the equations

  • equation image

which equations are only a part of the complete Maxwell's equations and therefore the solutions of (16), that are basic modes, are not the solutions of the Maxwell's equations. If we use similar reformulation steps which were used during the reformulation process from (11) to (14) with the application of (12)), because the aik terms appear only out of the derivations in (16), it follows from (16)) that

  • equation image

where the index m means of the inhomogeneous basic modes and m = 1, 2, … M, where M is the total number of the inhomogeneous basic modes, i.e., the total number of the possible (independent) solutions of (17). The number M depends on the given task, i.e., it depends on the number of the K′ comoving systems in the given task and the propagation situation in the independent K′ systems. However, every possible solution of (17), i.e., every basic mode must be taken into account in the following.

[19] It is clear that the inhomogeneous basic modes exist in given points in K′ (or in equivalent points in K) and the completeness of the Maxwell's equations guarantees that in every point (or surface, etc.) we shall have only such number of independent inhomogeneous basic modes which is necessary to derive a complete solution. (This is a checking possibility of the correct solution, too. See more about this question in classic applications [e.g., Ferencz et al., 2001].)

[20] It is important that it is necessary to solve the Maxwell's equations (together with the constitutive equations, applying the known methods valid in resting media) only in the comoving K′ systems, if we derive the inhomogeneous basic modes. The constitutive equations in the K′ systems are well known, because these are the constitutive relations in a resting, nonmoving medium since the medium is resting in K′. Therefore the application of the MIBM resulted in first step a well-handling mode of the problem, because in the derivation of the medium-signal interactions is very problematic in the K laboratory coordinate system if a complex medium is moving in it, however, we have known methods to solve these relations in resting media.

[21] The current density

  • equation image

where σr source(xs) is the source of the propagating signal and σr medium(xs) contains the effects of every particle, i.e., charge motions and elementary circular currents, too.

[22] The equation system (17) is valid in the comoving K′ systems and therefore we can solve this equation system as we know from the classic electrodynamics [see, e.g., Kong, 1975; Ferencz et al., 2001; Ferencz, 2004, 2005].

3.4. The Coupling Equations

[23] Knowing the derived inhomogeneous basic modes as it was presented in section 3.3, we can say that the complete solution of the Maxwell's equations is

  • equation image

Here we do not apply the Einstein's summation in the case of the index m in Frs m, therefore appear here the m as a separate index in the classic Σ symbol. We know that the equations (16) are fulfilled for every Frs m independently on other modes and therefore the members which are present in (16) will disappear from (15). However, the key parameters of these Frs m modes are not fixed to the excitation signal or to each other, i.e., they are unknown. These parameters will appear after the transformation of these Frs m from K′ to K in Fik and the determination of these parameters is possible by solving the remaining part of (15) (ari are known, they are determined by the motion velocity field). Applying (19) in (15) the final form of (15) will be

  • equation image

We call (20) the “coupling equations.” By solving (20) we shall have the complete full-wave solution of the problem as it is detectable in the laboratory system K, i.e., comoving with the observer (i.e., space equipment tracking station, laboratory, etc.).

[24] This solution method is valid for monochromatic and UWB signals, and for isotropic, anisotropic, bianisotropic media, too, because no restrictions were used in the form of the excitation or of the propagating signal or in the form of the medium-signal interactions.

4. Solution of the Maxwell's Equations by MIBM Using Equation (5)

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References

[25] In some cases it is not convenient to incorporate every medium-signal interactions into the four-dimensional current density (i.e., field induced currents, circular currents, polarization effects, etc., together with the excitation, source signal). In these cases it is better to start by using (5). The general structure of the propagation is unchanged, it is given in Figures 1 and 2 and however, the value of σi current density differs from the value of σi appearing in (7). Therefore

  • equation image
  • equation image

where

  • equation image

Applying (22) in system K, the form of the Maxwell's equations will be

  • equation image

[26] In the homogeneous case, when the medium is homogeneous and time-invariant in the coordinate system K′, and the moving velocity of the medium is homogeneous, i.e., uniform and time-invariant—using the derivation presented in section 3.1—we obtain the forms of equations (23) as

  • equation image

and from (24) by using (12), etc., it follows that

  • equation image

in the K′ system in these cases. We will use the equations (24) in the definition of the inhomogeneous basic modes, as it was made in section 3.3, too.

4.1. The General (Inhomogeneous) Case Using Equation (5)

[27] In the same manner as we showed earlier in section 3.2 it is possible to derive from (23) that

  • equation image

Let us define the inhomogeneous basic modes in a similar manner as it was done in section 3.3. In this case this means that the inhomogeneous basic modes will be the solutions of the equation system

  • equation image

i.e., the inhomogeneous basic modes from m = 1 to m = M are the solutions of

  • equation image

valid in the K′ comoving coordinate systems. Therefore it is not necessary to know or to derive the Fik [LEFT RIGHT ARROW] Gik constitutive equations or relations in the K laboratory system (in the system of the observer), it is enough to know these relations in the comoving systems K′, which means the application of the classic forms knowing in resting media.

[28] Knowing the inhomogeneous basic modes derived by (28) in K′, it is possible to say that the complete solutions of the Maxwell's equations are

  • equation image

Therefore the coupling equations in this case, derived by identical manner as it was done in section 3.4, are the following

  • equation image

[29] It is clear that in relativistic formalism the definition of the permittivity, permeability, conductivity, etc., is very complicate [Marx, 1952, 1953] even in the cases of a single, strictly monochromatic signal except in the K′ system comoving with the medium, and it is not possible to define these relations in the cases of the UWB or general shaped signals even in comoving systems too [Ferencz et al., 2001]. However, using this method it is enough to know these relations in the systems comoving with the media or to use the derivation methods applicable in K′ system to derive the electromagnetic field components, as it is clear from (20) and (30). If we know the Fik [LEFT RIGHT ARROW] Gik relations, it is possible to solve (17) or (28), and the solution of the coupling equations (20) or (30) produces the solution of the problem, and it is not necessary to know or to use any hypothetical or well derived, but very complicated Fik [LEFT RIGHT ARROW] Gik relations (see, e.g., Marx [1953] for an example of these very complicated relations).

5. The Form of the Coupling Equations in Stratified Velocity Fields

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References

[30] If the velocity field is stratified, i.e., the velocity field contains layers only with uniform velocity inside the layers, then it is possible to describe this velocity field by applying the Heaviside distribution (or in other words gate functions or step functions). The sketch of the situation is given in Figure 3. (This velocity field model is in a complete harmony with the general idea of the special relativity, because in this case we have definite layers of medium which are moving with constant velocities as it is necessary to have exact inertia systems comoving with these medium layers.) Let the Heaviside distribution be 1(x) and we will use the symbols of Figure 3 and the equations (1) and (4). In this case the velocity field is

  • equation image

where

  • equation image

(Here we do not apply the Einstein's summation in the case of the index j in sj (xq) and in jvi, therefore appear here the j as a separate index in the classic Σ symbol.) Therefore

  • equation image

where jaik0 are the components of the Lorentz tensor inside the jth layer, i.e., calculating by jvi, and the index j on the front side of the symbols are the number of the jth layer.

image

Figure 3. The sketch of the medium/velocity field model if we describe the stratified velocity field of the inhomogeneous media by Heaviside distributions. Here j are the number of the layers with uniform velocity inside the given layer, Aj are the boundary surfaces between the layers, jvi are the velocity of the individual layers, and x(Aj)i are the coordinates of the Aj surfaces.

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[31] In the coupling equations (20) or (30) the (ariask) elements are present. These elements are

  • equation image

where

  • equation image

as a consequence of (32). Therefore

  • equation image

Using (35) it is possible to derive the elements of (20) or (30). In first step let us see the form of the individual members in (20) or (30). These are

  • equation image

which are

  • equation image

where the derivatives of sj(xq) are Dirac-delta operators. After a few trivial mathematical steps we can derive from (36) that

  • equation image

Applying these results in (20) or (30) we obtain the following:

  • equation image

or

  • equation image

6. The Form of the Coupling Equations in Continuously Varying Velocity Field

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References

[32] In this case the velocity of the inhomogeneous medium is changing in space continuously, i.e., the derivatives of aik are limited, the Dirac-delta operator will not appear between the derivatives of aik. However, it is important that this velocity field, which is changing in space, is constant in time and therefore the criteria of the inertia systems is valid for the K′ systems, comoving in every point with this media. Therefore the form of the individual members in (20) or (30) will be the following:

  • equation image

where ari−1 and ask−1 are the inverses of ari and ask. Thus

  • equation image

Therefore the form of the coupling equations (30) is

  • equation image

[33] It was clarified in section 3.3 that the inhomogeneous basic modes exist in given points in K′ (or in equivalent points in K) and the completeness of the Maxwell's equations guarantees that in every point (or surface, etc.) we shall have only such number of independent inhomogeneous basic modes which is necessary to derive a complete solution. Therefore we shall have a point-by-point changing set of inhomogeneous basic modes with enough and only enough free parameters in every points in the coupling equations and the coupling equations guarantee through the coupling (and fixing) these free parameters that the (dispersive or nondispersive, inhomogeneous) medium-signal interaction will happen with the real (correct) parameters (e.g., frequency) in every points, in every K′ systems.

7. An Example of Application

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References

[34] Let the application example be the propagation in the moving (windy) troposphere, described by Birkemeier et al. [1968] and later, e.g., by Ferencz [1980b]. This is a very well measured and widely investigated propagation case, and beside these the propagation geometry is simple, therefore it is a good demonstration of the practical applications. (It was investigated earlier by scattering theory and by relativistic ray tracing.) We will use MKSA system in this section, because it is better suited for practical example, however, it is important to remember that the coupling equations in the given form are valid in cgs. The simplified sketch of the propagation in a stratified troposphere model is given in Figure 4. In this case the model has two layers. The first layer (circled numeral 1) is the reference, i.e., this (ɛ0, μ0) medium is resting in laboratory system K, and the second layer is moving with equation image2 = v · equation image1 velocity, where v = constant. In this simple model the medium 1 is homogeneous; however, the medium 2 is inhomogeneous. Let the source signal be monochromatic.

image

Figure 4. The sketch of the propagation in the example of application.

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[35] The steps of the derivation of the solution are as follows:

[36] 1. Derivation of all inhomogeneous basic modes by applying (28) in the comoving coordinate systems, which have the most optimal orientation to the solution of (28) in the given case of the boundary surface A2 and the propagation through medium 2. These are well known derivations in the classic wave theory.

[37] 2. Transformation of the derived inhomogeneous basic modes into the comoving systems K′ and K, respectively. After these rotations of coordinate systems the forms of the derived inhomogeneous basic modes will be compatible with the assumptions used during the derivation of the coupling equations (30), or in this case (39). (The rotations of the coordinate systems are also trivial.)

[38] 3. In last step it is possible to solve the coupling equations (39) and to derive the received wave parameters, e.g., the received signal frequency in this case of application [see, e.g., Birkemeier at al., 1968].

[39] Therefore in first step we can write the solution of (28) as it is well known, in homogeneous parts of the layers in the form

  • equation image

in a comoving system, which is fitted to the propagation plane wave in well known manner. (Here E0 and H0 are the electric and magnetic amplitudes of the electromagnetic field, respectively, ω is the angular frequency, equation image is the three-dimensional propagation factor, ξ is the x-component of the propagation factor, and Z0 is the wave impedance of the medium. In the inhomogeneous parts of the individual layers we can derive the solutions by the method given by Ferencz [2004] for monochromatic waves. A consequence of the earlier steps is the fact, that it is necessary to rotate these comoving systems into the comoving systems K′ which are defined in Figures 1 and 2. In (42)

  • equation image

therefore

  • equation image

and

  • equation image

where c is the velocity of light in vacuum, Z00 is the wave impedance of the vacuum and n is the refraction index in comoving system. Because (42) is written in MKSA unit system, therefore to the application of these results in (3), (5) or (7) and therefore in the coupling equations, it is necessary to make a well known unit transformation, i.e.

  • equation image

in our example, where the index MKSA means the values derived by equations valid in MKSA unit system.

[40] In the given model (Figure 4) the Dirac-delta operators in the coupling equations produce two groups of coupling equations (39). The first group is connected to the excitation signal coming from the transmitter to the surface A2 and the second group is connected to the signal propagating after the traversing of the inhomogeneity in layer 2 and propagating toward the A2 (and after crossing A2 toward the receiver).

[41] Looking at the structure of the equation system (39) or (38) it is clear that in monochromatic cases this equation structure guarantees the phase invariance [e.g., Synge, 1965; Schmutzer, 1968] of the complete wave pattern. In the measurements of Birkemeier et al. [1968] the goal was the investigation of the frequency shift of the monochromatic signal during the propagation. To the derivation of the complete frequency shift in this simple model (in our case) it is enough to derive the frequency shift of the signal reaching the receiver (Figure 4) from the complete phase pattern of the inhomogeneous basic modes coupled by the equations (39) in the model given in Figure 4. The result of this derivation is

  • equation image

where ω0 = 2πf0, and f0 is the excitation signal frequency at the transmitter, which was 810 MHz and 960 MHz in the given experiments depending on the direction of the transmission between the two end of the transmitting path, κ ≅ k0ncosα1, ϕ0 is the antenna pointing angle given in Figure 4, similarly Θ0 and α1 are also given in Figure 4; and we can done usual simplifications in the calculations of the angles in K′ systems because in this case the wind velocity is vc. Beside these

  • equation image

Using the assumptions that vc and computing the Taylor series of (45), the relative frequency shift is computable and it is e.g., in third order approximation in the Taylor series

  • equation image

where C1, C3, etc. are constants and α1 is small. The model (46) is usable in the interpretation of measured data presented by Birkemeier et al. [1968] (see also Figures 5 and 6). These data were interpreted by Birkemeier et al. [1968] using a correct signal scattering model in which the signal is scattered by moving particles. But the model (46) describes the signal propagation through this moving media, too. The equation (46) contains only the odd members of the Taylor series, which is compatible with the measured Δf [LEFT RIGHT ARROW] ϕ0 curves. See Birkemeier et al. [1968] and Ferencz [1980b] for more.

image

Figure 5. Results of the frequency shift measurements during tropospheric propagation in 1966 [Birkemeier et al., 1968].

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image

Figure 6. Results of the frequency shift measurements during tropospheric propagation in 1967 [Birkemeier et al., 1968].

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[42] However, in (46) a new member appears, a general frequency shift which is independent of the antenna pointing angle ϕ0, but it depends on α1 elevation. In these measurements with the usual wind velocities this α1n · v/c factor is very small, but it is identifiable in more measured curves in the very small shifting of the frequency above the Δf = 0 line at the ϕ0 = 0 value. The wind velocities were in the 1 m/s ≤ v ≤ 15 m/s region, the refraction coefficient of the inhomogeneities of the troposphere along the propagation path was around 1.2, i.e., 1.05 < n < 2, and α1 of the antenna was maximum a few grads (applying a slightly curved Earth surface contour), remains below 20°, i.e., in 0.017 ≤ α1 ≤ 0.25 domain in radians. Using these parameters in (46) this additional frequency shift is very small, it is a Δf < 0.05 ∼ 0.1 Hz shifting of the frequency curves above the Δf = 0 line at the ϕ0 = 0. (Using this value in (46) it is possible to produce an estimation of the measured data (Figure 7), as an illustration of practical possibilities.)

image

Figure 7. Results of the frequency shift measurements during tropospheric propagation in 1967 [Birkemeier et al., 1968] and the application of (46); where plus signs are the computed data for the 10:10 to 11:15 CST measurement, crosses are the computed data for the 11:15 to 12:00 CST measurement, and squares are the computed data for the 12:00 to 13:15 CST measurement.

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8. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References

[43] With the application of the MIBM, it is possible to derive a complete and exact full wave solution of Maxwell's equations in moving, inhomogeneous media. It is an important fact, that it is not necessary to know or to derive the Fik [LEFT RIGHT ARROW] Gik relations in the K laboratory system, however, it is enough to know these relations in the K′ comoving systems, i.e., it is enough to know and to apply the classic Fik [LEFT RIGHT ARROW] Gik forms of these relations.

[44] The method is applicable for monochromatic, quasi-monochromatic and for general shaped UWB signals, and for simple and complex (anisotropic, bianisotropic, etc.) media.

[45] These solutions are valid if the velocity field of the media is constant; therefore the coordinate systems fixed to the observer or to the moving media are nonaccelerating, inertia systems, i.e., inside the validity of the special relativity. In the method we did not used any restrictions of the value of the velocity in the 0 < v < c range.

[46] It seems to be necessary a further detailed investigation with the comparison of the models using stratified velocity field description and using continuously varying velocity field description. The form of the derivatives appearing in the case in which the velocity field varies continuously in space has similar character as the derivatives appeared in the case of the general relativity.

[47] It is necessary to generalize the solution method to the cases appearing inside the validity of the general relativity.

Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References

[48] The first version of MIBM for monochromatic waves was presented by Ferencz [1978c]. In homogeneous media the propagating monochromatic signal can be described by a plane wave (or by waves of coordinate surfaces), and the phase function, further the propagation vector are defined. If the linear medium contains spatial inhomogeneity—which does not fit into the geometry of the wave pattern—the effect of reflection, scattering or diffraction will arise and the amplitude will change in space. Furthermore, as inhomogeneity means anisotropy as well, the signal will be a superposition of several signals, or – in other words – the signal will disintegrate into n modes (e.g., propagating and reflected, EM and HM, etc.). Now, let it be assumed for the presentation of the general nature of the method, that the signal is monochromatic, i.e., exp j(ωt − ϕ) or its combination. However, the method of inhomogeneous basic modes is valid in the case of general signal shapes, and impulses (UWB) too (see Ferencz et al. [2001] and Appendix B).

[49] Further, let a linear, time-invariant and bianisotropic medium be considered for demonstrating the method of inhomogeneous basic modes

  • equation image

The assumed form of the solution looked for is

  • equation image

where ai(equation image) are general space-depending envelope functions.

[50] Another possible form of the solution is as follows

  • equation image

where index i denotes the component solutions (modes), l means the real (v) and the imaginary (k) components

  • equation image

Executing some mathematical operations on Maxwell's equations, the following equation system can be obtained

  • equation image
  • equation image

where

  • equation image

Let the generalized propagation vector equation imagei = equation imageequation imagei be defined, as if that were the propagation vector of the quasi-homogeneous solution, so let equation imagei be the solution of the following dispersion equation

  • equation image

The so-called inhomogeneous basic modes are the fields belonging to equation imagei, which fulfil (A8)

  • equation image

Substituting these inhomogeneous basic modes into (A5), the remaining unknown ail (equation image) “combining functions” can be obtained from

  • equation image

and

  • equation image

Appendix B:: Maxwell's Equations in the Sense of Distributions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References

[51] Now let a medium be considered in which the medium parameters suddenly change at some open or closed Am surfaces having no intersections (Figure B1). The medium parameters within the volumes Vm are described by continuous functions joining each other with jumps at the boundary surfaces. Further, it is assumed that exp j(ωt − ϕ) type of the solutions can exist in volumes Vm. In such cases, the medium is characterized by distributions (functional). The form of the solution looked for in Vm is the following

  • equation image
image

Figure B1. The structure of the medium.

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[52] Furthermore, let the 1(x) Heaviside (or unit step) and the δ (x) Dirac-delta distributions be introduced. 1[equation image (pm, qm)] denotes the distribution, the value of that varies from 0 to 1 at the surface equation image = equation image(pm, qm). The vector equation image(pm, qm) represents the parametric equation of the surface Am. The different unit step functions 1[equation image (pm, qm)] belonging to the regular Am surfaces are defined so that it is possible to create the following distribution from them

  • equation image

which has 1 value in the region between surfaces Am−1 and Am, while it has 0 value everywhere out of that (see Figure B2).

image

Figure B2. To the definition of the distribution functions.

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[53] Considering that the derivative with generalized meaning of 1(x) is δ(x), the derivative of sm (equation image) can be obtained. This differs from 0 only at Am and Am−1 boundary surfaces

  • equation image

where equation image0m is definitely the normal vector of Am surface, points outward (to Am+1 in the present case).

[54] The complete solution to be determined can be obtained by the MIBM. Each mode is defined and determined according to (B1) in the sm (equation image) = 1 regions separately, and the whole solution can be built the following superposition

  • equation image

where M is the number of the continuous Vm regions.

[55] In the simplest case, when the medium parameters are homogeneous in each domain Vm, all the modes possibly existing in the domains derive from

  • equation image
  • equation image

equations. The form of the full solution is the one presented in (B4). Substituting this into Maxwell's equations, they can be decomposed into two groups. One group is valid within the regions, the solution of which is known from (B5) and (B6). The other is valid at the boundary surfaces, delivering the connection among the regions; this is called coupling equation system

  • equation image

Solving (B7), the complete solution can be obtained.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Initial Conditions and Formalism for Derivation
  5. 3. Solution of the Maxwell's Equations by MIBM Using
  6. 4. Solution of the Maxwell's Equations by MIBM Using
  7. 5. The Form of the Coupling Equations in Stratified Velocity Fields
  8. 6. The Form of the Coupling Equations in Continuously Varying Velocity Field
  9. 7. An Example of Application
  10. 8. Conclusions
  11. Appendix A:: Method of Inhomogeneous Basic Modes (MIBM)
  12. Appendix B:: Maxwell's Equations in the Sense of Distributions
  13. References
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