What is longitudinal wave motion
Gerhard W. Bruhn, Department of Mathematics at TU Darmstadt
Are there electrical scalar or longitudinal waves?
K. Meyl has expressed the view in various publications that in addition to the well-known transverse electromagnetic waves there are electrical longitudinal waves, which he calls scalar waves, characterized by the condition that the field is free of rotation E. (Part 1). Meyl relies on the wave equation of the electric field vector E..
However, it is known that the wave equation of the electric field vector E. has no direct experimental justification (Section 2), it is only a consequence of the actual basic equations of electrodynamics, the Maxwell equations, which since Faraday and Maxwell have had a comprehensive experimental basis that has been confirmed by practice. Not every solution of the wave equation (the electric field vector E.) leads back to a solution of the Maxwell equations, there are "ghost solutions" of the wave equation to which no solution of the Maxwell equations corresponds. Therefore, for solutions to the wave equation is a Insertion sample absolutely necessary in the Maxwell equations.
Meyl claims that longitudinal electric waves exist in air and even in a vacuum, generally in charge-free homogeneous media. Because the charges (with constant dielectric constant) are the sources of the electric field vector, freedom from charge also means freedom from source. Accordingly, in the case of longitudinal waves, the electric field vector must be free of rotation and swelling at the same time. However, one can easily see (Section 3) that the wave equation in this case is reduced to the disappearance of the 2nd time derivative of the field vector. This allows only a very simple type of solution that cannot describe any oscillation processes. Therefore, in charge-free homogeneous media (such as normal air or vacuum) no scalar waves exist.In other words: The scalar wave solutions of the wave equation constructed by Meyl do not pass the test in the Maxwell equations.
In the following section 4, on the basis of Maxwell's equations, the more general question is discussed whether the waiver of the freedom from charges enables longitudinal waves. It turns out first of all that the magnetic field H constant over time must stay. A possible scalar wave would be purely electric. In return, however, it follows (with the addition of Ohm's law for the charge movement) that electrical oscillations are also not possible in spite of moving charge distributions: Scalar waves do not exist.
The actual Reason for the non-existence of scalar waves under the terms of Sections 3 and 4 is that the Longitudinal demand Too great a restriction for the solutions of Maxwell's equations means because changes in the magnetic field over time and, in particular, oscillation processes of the magnetic field are directly suppressed, which then also prevents oscillations of the electric field.
Section 5 deals with the impossibility of faster than light solutions of the homogeneous wave equation.
Section 6 points out a persistent error in K. Meyl's publications, which transforms a homogeneous wave equation into an inhomogeneous wave equation.
Longitudinal waves are waves in which the direction of oscillation (of the oscillating particles, the oscillation vector) coincides with the direction of propagation. In contrast to this, with transverse waves the oscillation takes place perpendicular to the direction of propagation. Both types of waves are possible in elastomechanics. According to Sommerfeld , p.102, applies to elastic displacements s(x, t) the differential equation
(1.1)sdd = α degrees div s + β Δs with positive constants α, β,
what withdegree div s = Δs + red red s also in the form
(1.1')sdd= (α + β) degrees div s- red red s= (α + β) Δs + β red red s,
can be written. If you look at the trivial case sdd= 0 excludes, at least one of the sizes div sor red s not equal to zero. Solutions of (1.1) with
(1.2) red s = 0
become Compression waves called while one in the case
(1.3) various s = 0
of Shear waves or Torsional waves speaks. When specializing in level Waves are compression waves always longitudinal while level Shear waves always oscillate transversely, see , p.172. Note that setting the Direction of propagation for non-plane waves can be problematic.
For compression waves one obtains the wave equation from (1.1 ')
(1.4)sdd = (α + β) Δs,
while (1.1) the wave equation for shear waves
(1.5)sdd = β Δs
2. Shear waves in electrodynamics
The electrodynamics is in Case of a charge-free and current-free homogeneous medium (with constant material coefficients ε and μ) from the (homogeneous) wave equation
(2.1) ΔE = degree div E.- Red Red E. = c-2 E.dd(with the speed of light c; c-2 = εμ).
controlled. It should be noted, however, that the wave equation (2.1) no basic equation of electrodynamics, but rather episode from the (homogeneous) Maxwell equations of the electromagnetic field (E.,H)
(2.2 a, b) red E = - μHt, div E. = 0,
(2.3 a, b) red H = + εE.t, div H = 0.
The literature occasionally (and apparently not without reason) warns against this misunderstanding of the wave equation (2.1). For example, F. Hund writes in , p. 241 f. In 1946:
"The equations (2)
This means that for the case of a charge and current-free homogeneous medium to the wave equation (2.1) for E. always add the equation (2.2 b), div E. = 0, because div D. = ε div E. = ρ is the electrical charge density. Thus, the solutions of (2.1) sought for electrodynamics are always "shear waves" or transverse waves in the sense of the elasto-mechanics of section 1.
3. K. Meyl's scalar waves in neutral media
K. Meyl makes two contradicting statements about the medium of his scalar waves: On the one hand, he writes at the beginning of the section Vortex model from , analogously also in :
"Such longitudinal waves obviously also exist without plasma in the air and even in vacuum."
It follows that a neutral Medium, such as vacuum or (non-ionized) air, or in general a plasma-free The dielectric is to be taken as a basis, i.e. the charge density ρ = εdiv E. (Equation (3) in Meyls Fig. 2, see below), so also div E. itself, is to be assumed to be zero.
On the other hand is found in Meyls picture 2 the condition "Div E. ≠ 0".
We first want to consider the important case of the charge-free media such as vacuum and air and postpone the discussion of the second condition, which is not fulfilled in neutral media, to the following section.
K. Meyl has commented on scalar waves on various occasions, e.g. in  and in . In  it says in the signature to picture 2
"picture 2: The scalar part of the wave equation describes longitudinal electrical waves (derivation of plasma waves). "
The terms “scalar” and “longitudinal” are identified here. In addition, a longitudinal wave becomes red in the sense of the definition in section 1 as a solution of the wave equation (2.1) under the additional condition E = 0 Are defined.
From the equation
(2.1) Δ E = degree div E.- Red Red E. = c-2 E.dd
one can see immediately that, assuming no charge for longitudinal waves, the already mentioned trivial case with div E. = 0 and red E = 0 is present, the
E.dd = c2(grad div E. - Red Red E.)= 0
and the type of solution
(3.1)E.(x, t) = E.0(x) + t E.1(x)
entails. These solutions cannot be called waves, they cannot be described no oscillation.
Scalar waves cannot exist in a neutral homogeneous medium.
This also follows from Meyl's calculation in picture 2 (without "Div E. ≠ 0 "to be observed): From ρ / ε = div E. = 0 with E. = - degree φ direct
Δφ = div degrees φ = 0,
thus, according to the equation to the right of "plasma wave", also φdd = 0, which, analogous to (3.1), only offers solutions of the form
(3.2) φ (x, t) = φ0(x) + t φ1(x) with Δφk = 0
who, as above, no wave represent. (We will talk about an error contained in the "plasma wave equation" (but ineffective here) in Section 6.)
The result achieved can also be expressed differently: The scalar wave solutions of the wave equation constructed by K. Meyl have the property div E. ≠ 0 and therefore do not pass the test in the Maxwell equations, where div E. = 0 is required.
4. Scalar waves at charge density ρ ≠ 0?
We now want to face the compulsion of the condition div E. = 0 evade by
(2.2 b ') ε div E.= ρ
with charge density ρ ≠ 0 instead of the homogeneous Maxwell equation (2.2 b).
Because of the warning from F. Hund cited in Section 2 (, pp.241 f.), We start from the Maxwell equations (2.2-3) modified by (2.2 b '). Existing charges can move and thus generate a current density j, which in turn makes a contribution to the rotation of the magnetic field. Therefore Maxwell's equation (2.3 a) must also be modified to
(2.3 a ') red H = j + εE.t
As a consequence of the longitudinal condition red E = 0 one obtains according to (2.2 a) Ht=0,So a constant over time Magnetic field H(x). I.e. any scalar wave would be pure electrical Wave.
Furthermore, the time differentiation of (2.3 a ') gives
(4.2) (j + εE.t)t = 0,
what the temporal constancy of the sum j + εE.t, i.e. of charge and dielectric displacement current density.
In order to come to a result with regard to possible waves, we want Ohm's law, i.e. a linear relationship between j and electric field E., accept as valid with constant conductivity σ> 0,
(4.3)j = σ E..
The combination of equations (4.2-3) yields the linear ordinary differential equation of the 2nd order
(4.4) (σE.+ εE.t)t = 0
whose general solution can be written down as closed:
(4.5)E.(x, t) = E0(x) + σ-1j0 (x) (1 - e- σ / ε t )
It is E.0the initial state of the electric field for t = 0. Furthermore is j0 = ε E.t|t = 0, the initially flowing dielectric displacement current D.t, which decays exponentially for increasing t according to (4.5),
(4.6)D.t = εE.t = j0 e- σ / ε t.
Equation (4.5) describes an exponential adaptation process of the field E. from the initial state E.0to the final state that occurs after the displacement currents have subsided (for large t) E.0+ σ-1j0. From an oscillation process of E. can not be a question:
Even the admission of mobile charge density does not allow scalar waves.
The deeper one Reason for the non-existence of scalar waves under the conditions of sections 3 and 4 is that the longitudinal condition is red E = 0 too strong a restriction for the solutions of Maxwell's equations means, because initially oscillation processes of the magnetic field caused by (2.2 a) are also red E = 0can be suppressed directly. But the Maxwell equation (2.3 a ') then enables red with a temporally constant left-hand side H for the electric field occurring on the right E. no more vibration solution.
additive After dualization of the Maxwell equations, K. Meyl set up a so-called fundamental field equation, whereby the electric and magnetic field are both as source-free are assumed. The fact that the electric field is free of sources contradicts that in Meyls picture 2 assumption made from  for scalar waves div E.≠ 0. Nevertheless, the question arises whether the dualization of Maxwell's equations can lead to the existence of scalar waves. This question is discussed in  independently of the question of the sources of E. answered negatively: The dualization of Maxwell's equations also does not allow scalar waves.
5. Superluminal solutions of the wave equation?
Under No. 4 of the introductory part of , K. Meyl announces the
"Transmission of scalar waves at 1.5 times the speed of light!"
at. Now this statement is obsolete after the previous section: Because there are no scalar waves, there cannot of course be any waves with faster than light speeds.
Still, it's an interesting question as to whether the wave equation
(5.1) ΔE. = c-2 E.dd(with the speed of light c).
Can have solutions that propagate faster than light. You have to know that there are different speed terms for waves, e.g. the phase speed. It is a triviality that by superimposing sine waves of almost the same frequency one can generate waves with any superlight phase velocities. However, the hard proposition of mathematics about the wave equation relates to the Signal speed. It is:
The wave equation (5.1) has no solutions with a signal velocity> c.
Whenever someone has effects or particles (such as the fabulous tachions) signalspeed> c should discover:
These effects cannot be caused by the spread of solutions Wave equation be caused. That, and no more, says mathematics. So if K. Meyl is the
"Transmission of scalar waves at 1.5 times the speed of light!"
announces, it may be what will be behind it, but not solutions of the wave equation with signal speed> c.
Mathematics does not prohibit tachions in general, but only those that satisfy the wave equation.
It remains to explain what is belowSignal speed is to be understood:
Under a E.-signal we understand E.Field, which at the start time t = 0 outside a sphere of radius R0> 0 is identical to zero. The statement of math for E.-Signal solutions of the wave equation then says that for t> 0 the initial state E = 0 still outside the (concentric) sphere of radius R0+ ct has been preserved. The E.-signal has for t> 0 (at most) the sphere of radius R0+ ct fulfilled, the signal sphere radius has changed with the Signal speed c enlarged.
The proof is a well-known result from the theory of partial differential equations, which is elementary in principle, but a little tricky to carry out. The reader is referred to, for example, , p.379 ff. The proof uses the Gaussian integral theorem. There is well understandable evidence for the spatially one-dimensional case.
6. An inhomogeneous wave equation in K. Meyl
The statement just discussed about the impossibility of a signal speed> c relates to the homogeneous Wave equation. No such simple statement is possible for the inhomogeneous wave equation, because the inhomogeneity, the "right side", could "import" faster than light.
In picture 2 from  and  (see our section 3) there is a to the right of "plasma wave" inhomogeneous Wave equation, the inhomogeneity of which is expressly pointed out in the accompanying text. This inhomogeneity isfaulty: In the equation, there is no equal sign at the point marked with a red arrow, as can be confirmed immediately by recalculating. After correction, the homogeneous Wave equation for φ, the expression -ρ / ε is identical to Δφ. But as explained in the previous sections: This equation is obsolete anyway.
 K. Meyl, longitudinal wave experiment according to Nikola Tesla,
 K. Meyl, Scalar Waves: Theory and Experiments, Journal of Scientific Exploration, Vol.15, No.2, pp. 199-205, 2001
 A.Sommerfeld, Mechanics of Deformable Media, Akademische Verlagsgesellschaft Leipzig, 1949
 G. Joos, Textbook of Theoretical Physics, 15th edition, AULA-Verlag, 1989
 F. Hund, Theory of Electricity and Magnetism, Bibliographisches Institut Leipzig, 1947
 R. Courant - D. Hilbert, Methods of Mathematical Physics II, 2nd edition, Springer-Verlag, 1937/1968
 G.W. Bruhn, On the Existence of K. Meyl’s Scalar Waves, Journal of Scientific Exploration, Vol.15, No.2, pp. 206-210, 2001
 G.W. Bruhn, Do K. Meyl's Scalar Waves Exist?
 G.W. Bruhn, K. Meyls Fundamental Field Equation,
 Experimental reports on Meyl's scalar waves:
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