Analysis of the Doppler Effect Based on the Full Maxwell Equations
YEVGEN V. CHESNOKOV
Institute of Cybernetics of the Ukrainian Acad. Sci.,
02091 Trostyanetska str. 12, apt.155 Kyiv
UKRAINE
IVAN V. KAZACHKOV
Department of Information Technology and Data Analysis
Nizhyn Mykola Gogol State University
16600 Grafska str. 2, Nizhyn
UKRAINE
Abstract: - In the previous paper, a modification of Maxwell's equations was proposed, from which formula for
Doppler effect follows. However, as was noted later, the equations proposed do not have symmetry with respect
to the transformation B→-E, E→B, which the original Maxwell equations have and which was discovered by
Heaviside in 1893. The equations proposed in present paper have this symmetry. The obtained equations are
analyzed for several physical situations.
Key-Words: - Maxwell Equations; Doppler Effect; Symmetry of Equations
Received: August 24, 2021. Revised: April 15, 2022. Accepted: May 17, 2022. Published: July 2, 2022.
1 Introduction to the Problem
In the paper [1], the formula for the Doppler effect
was given, which describes the dependence of the
radiation frequency recorded by the observer,
depending on the angle between the direction to the
source and the direction of movement of the source.
As shown in our work [2], the expression for the
Doppler effect can be written as
(1)
which is just a record of the cosine theorem for the
difference between the velocity vectors of the wave
front and the source. Here k is the wave vector, Ω -
frequency, - vector of movement of the source of
light, c - the speed of light.
In the same place, a modification of Maxwell's
equations was proposed, from which formula (1)
follows. However, as was noted later, the equations
proposed in [2] do not have symmetry with respect to
the transformation
, ,
which the original Maxwell equations have and
discovered by Heaviside in 1893 (see [3]). The
equations proposed below have this kind symmetry
as shown in the present paper.
2 Mathematical model
2.1 The Modified Maxwell Equations
Consider the modified Maxwell equations of the
following form:
,
(2)
.
Here and further, the Heaviside system of units is
used everywhere, according to which ћ = c = 1.
Here the time derivatives are total, i.e.,
d/dt=∂/∂t+(v∙), where v - constant speed of the
source.
2.2 Fourier Expansion for the Modified
Maxwell Equations
Since the equations are linear with constant
coefficients, we use the Fourier expansion to solve
them:
,
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DOI: 10.37394/232021.2022.2.16
Yevgen V. Chesnokov, Ivan V. Kazachkov
E-ISSN: 2732-9976
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Volume 2, 2022
,
.
In this representation referred to in the literature
as the impulse representation, system (2) has the
form:
,
(3)
where .
The determinant of the system (3) is
. (4)
By substitution
,
(5)
,
where are:
– vector,
– scalar
– pseudoscalar
potentials, we can get the following conditions for the
potentials:
, (6)
, (7)
. (8)
2.3 The Modified Maxwell Equations in
Coordinate Form
The above is presented in the coordinate form as
follows:
, (9)
,
where are
, (10)
, (11)
. (12)
Applying the operator
to
the calibration condition
, yields
.
And using the continuity equation
,
we obtain an expression for the charge density:
. (13)
It follows from the first formula (9) that the
divergence of the magnetic field is not equal to zero
in motion, while at rest it is still zero:
, (14)
therefore, the lines of force of a magnetic field source
moving at a constant speed are not closed circles in
classical electrodynamics.
2.4 The Galilean Transformations
Applying the formulae (5) – (8) yields:
,
(15)
,
or in coordinate form
. (16)
The expressions in (9):
(17)
obviously mean the transition to a moving frame of
reference (the Galilean transformations).
Thus, the scalar potential and vector potential
create a pair in these transformations. And а
pseudoscalar potential does not take part in this as
seen from direct calculation: the expression
is invariant under transformation (17).
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DOI: 10.37394/232021.2022.2.16
Yevgen V. Chesnokov, Ivan V. Kazachkov
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Volume 2, 2022
2.5 The Inverse Galilean Transformations
The inverse Galilean transformations have the form:
(18)
Unlike the Lorentz transformations used in
relativistic electrodynamics, formulas (17) and (18)
are asymmetric, since in formulas (17) the medium in
which the wave propagates is at rest, and in (18) it
moves relative to the coordinate system with a speed
of .
3 Change of Variables
The equations (15) suggest the idea to make a change
of variables:
(19)
In such variables the system (2) transforms to:
,
(20)
.
From the continuity equation
follows
,
(21)
4 Energy-Momentum Tensor and
Dispersion Equation
4.1 Energy-Momentum Tensor
Let us scalarly multiply the first of equations (20) by
the vector , and the second by the vector
, and add, we get
And then multiply vectorially (on the right) the first
of equations (20) by the vector , β, and the second
(on the left) by the vector ε and add, we get
.
4.2 Dispersion Equation
Dispersion equation for the system (20) has
the following form:
. (22)
The roots of this equation are:
(longitudinal mode),
(transversal modes).
Recalling the definition of s, we have:
(23)
and
. (24)
Denoting by θ the angle between the vectors and ,
we obtain the last expression in the form
,
or
. (25)
Here
– the amplitude of the phase speed for
transversal wave. This expression (25) was given in
[1] ( was accepted in (25)).
Formula (25) is actually a record of the cosine
theorem for the difference of vectors. Expressions for
the group velocity and the delayed Green's function
are given in [2].
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DOI: 10.37394/232021.2022.2.16
Yevgen V. Chesnokov, Ivan V. Kazachkov
E-ISSN: 2732-9976
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Volume 2, 2022
5 Conclusion
From the results presented above follows that the
non-relativistic explanation of the Doppler effect,
based on the concept of a continuous medium in
which elastic - longitudinal and transverse -
oscillations propagate, may explain some more about
the Doppler effect, as well as about the other
interesting features.
References:
[1] Ether (Part 6). Doppler Effect (In Russian)
https://youtu.be/x20e0R7y2es.
[2] Yevgen V. Chesnokov, Ivan V. Kazachkov.
(2019) About the Maxwell Equations for
Electromagnetic Field and Peculiarity Analysis of the
Wave Spreading// International Journal of Applied
Physics, 4, 14-21,
http://www.iaras.org/iaras/journals/ijap.
[3] Heaviside O. Some properties of the Maxwell
equations // Phil. Trans. Roy. Soc. (London) A. –
1893. – 183. – P. 423-430.
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This article is published under the terms of the Creative
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DOI: 10.37394/232021.2022.2.16
Yevgen V. Chesnokov, Ivan V. Kazachkov
E-ISSN: 2732-9976
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