Longitudinal polarization of electromagnetic waves is the result of imposing particular boundary conditions, e.g. in waveguides, that break  Lorentz invariance. There's no such notion as an electroscalar wave-it can't be given any meaning. While each component of the electric and magnetic fields of an electromagnetic wave satisfies the wave equation, which is the same as for any wave, such as sound or water, the constraints that the components must satisfy, which express the fact that they are electric and magnetic fields, solutions of Maxwell's equations, means that they are different from scalar waves.

All solutions of the wave equation don't satisfy Maxwell's equations; the latter describe electric and magnetic fields and only solutions of both sets describe electromagnetic waves. 

For a pictorial image of the EM field closely related to the Lienard Wiechert potentials see "The Electromagnetic Field in Space-time" at users.rcn.com/eslowry/elmag.htm  .    I too am skeptical of the "Longtitudinal ElectroScalar Wave".

The 'scalar waves' (if propagation of the scalar potential can be treated as the wave process) have been detected in the paper of Chubykalo et al.

http://arxiv.org/abs/physics/0010036

Also Monstein and Wesley published a paper on detection of the longitudinal EM waves but their experimental results are questionable. Despite some arguments of Wesley are reasonable.

The existence of "inseparable" EM potential four-vector is necessary from the point of view of the self-sufficient potential formalism in EM theory. So, the radiation of so-called "scalar potential", e.g., in the axial direction of the electric dipole is quite real. However, the detection of respective "Zero-Magnetic" (ZM) waves must be examined from the point of view of the quantum electrodynamics. The principal question is: does the Aharonov-Bohm effect exist if the interference path does not envelop the source of ZM waves? See Lochak's papers, for example. My opinion concerning those is much more skeptical now comparing with the same two years ago... :-(

I met with this kind of theories. All of them are based on the potentials, which are derived from Maxwell's equations. But the paradox is that Maxwell's equations easily implies the impossibility of the existence of longitudinal EM waves. So, the one who wants to prove and detect such waves must first write and prove new equations.

M. L. Shur: I'm reading right now a book by Thomas E. Bearden  and he alleges that when today talking about Maxwell's equations is not the original, he wrote “Maxwell’s” vector equations taught in university are actually Heaviside’s truncated equations, and are only a simplified version of what Maxwell originally wrote.http://www.cheniere.org/references/maxwell.htm. Is that true? If true perhaps that's "your new equation" ?

Indeed, what we call Maxwell's equations, is the equations that were written by Heaviside. Maxwell's equations themselves are much more complicated. I must admit  honestly.that I know only this  Heaviside's (.Maxwell's) system of equations.  I would be grateful if you would send them to me, or give a link..

M. L. Shur: I just bought it from http://www.amazon.com/A-Dynamical-Theory-Electromagnetic-Field/dp/1579100155.

Here some other links I found

http://rexresearch.com/maxwell.htm and http://www.cheniere.org/images/EMfndns1/MaxQuatrnssm.jpg

  Dear Mr. Lovhaug!

I am extremely grateful for links to Maxwell's book. I was always amazed and interested in how Geniuses think. I read the works of Boltzmann, Lorentz and others, and got great pleasure following their  arguments and reasoning .

   I should point out that the original Maxwell's equations identical with the Heaviside equations that we use today.

Sincerely

Thank you so much, Mr. Baumgarten.

Here's experimental validation that purely longitudinally polarized light exists. http://www.nature.com/nphoton/journal/v2/n8/full/nphoton.2008.127.html

I am skeptical about the idea of a EM longitudinal wave.  If you want a basis for visualizing classical EM fields I suggest  users.rcn.com/eslowry/elmag.htm   .

users.rcn.com/eslowry/elmag.htm

Edvard Lowry:  Thomas Wood gave us an link and said "Here's experimental validation that purely longitudinally polarized light exists. ".  Do you think somethings is wrong here since you say you are "skeptical about the idea of a EM longitudinal wave." ?

Longitudinal ElectroScalar Wave implies that energy is being transmitted without a momentum vector. This doesn't happen in the differential equations of Maxwell  when applied to an infinitesimal space and time.

On the macro scale familiar devices have been constructed and operated, in which a symmetry causes the integrated momentum transfer to be zero, while energy is transferred.

A simplest device to describe is a pair of concentric conducting spheres separated by an insulating medium and operated as a capacitor. There is a longitudinal electric field between the spheres that hardly anyone disputes. Infinitesimally there are Electric field vectors operating in the insulator causing longitudinal electric displacement, but all of these field vectors add to zero because of the symmetry. Notice that the magnetic field B in the insulator is zero by local interference. The device taken as a whole from outside is a scalar wave generator and receiver set, more correctly when the electrical connections are made through shielded coaxial cables and with AC power wave form. 

A search on RG for scalar wave brings more than a thousand articles and more than a hundred questions. Terminology Is not always defined clearly, leading to many objections. In the wider literature there is history of researchers adding something to Maxwell with various levels of success or failure. Central to all of the scalar wave versions is the requirement of a symmetry to allow energy to transfer without a momentum vector related to the Poynting method.

Some of the claims are speculative or in obvious error. One claim is that a new branch of physics is being discovered.  Another claim is the scalar wave moves in time but not in space, impossible for an open system, but fairly ordinary for the capacitor example.  

Jerry Decker: You write: "This doesn't happen in the differential equations of Maxwell". Do you mean the equations that were written by Heaviside. Maxwell's equations themselves are much more complicated.  

Maxwell's original 8 equations in their 20 variations are all vector equations. There is no way to transmit energy without momentum in vector equations except by an integrated symmetry where one vector cancels out another.

I started to read me up again reading James Clerk Maxwell, The dynamical theory of the electormagnetic field edited by Thomas F. Torrance once more. In the introduction I read about how Maxwell translated Faraday's idea into mahematical forms. Faraday, in his mind's eye, saw lines of force traversing all space where the mathematicans saw centres of force atracting at distance. I search for more information about Faraday and read Jerry Decker last comment about Maxvell's equations once more.

Maxwell's original 8 equations in their 20 variations are all vector equations. There is no way to transmit energy without momentum in vector equations except by an integrated symmetry where one vector cancels out another.

Bergyl can be that kind of symmetry and Bergyl is the same as Faraday's force.

So I want to thank you a lot Jerry Decker for your answer on my question. 

I would also be very grateful for all the input about the road I should now proceed with Bergyl.

https://www.researchgate.net/project/Bergyl

Sorry  Solveig Løvhaug,  I don't have additional advice, but would like to read what you develop  about it.

Look at many types of waves in mechanics: longitudunal, transverse, of some specific shape... Analogous situation should take place in luminiferous medium also. About one specific wave type in the medium, "magnetodynamic" near-field wave, one can read here:Article Magnetodynamic waves in the air

As far as I have been able to determine, it seems that a stationary point charge of changing intensity does not have an associated magnetic field because the conditions for the generation of a magnetic field have not been met ie. movement of the charge. The resulting expanding or contracting E-field is scalar in that it is a spherical and thus omnidirectional and can only be described in terms of its intensity.

To an observer that is stationary with respect to this pulsating charge, the wavefront presents itself longitudinally in a similar fashion to a point source of sound. (Langmuir wave ?)

I believe it was Eric Dollard who first started using the term magneto-dielectric to distinguish between processes that naturally generate transverse emanations (electromagnetic) as opposed to those that generate longitudinal ones. Looking at a parallel plate capacitor, a magnetic field circulates during charge and discharge, so conversely, a circulating magnetic field between two plates must result in a stationary electric field. Since this field is circulating in the dielectric, the term for this process as being magneto-dielectric seems appropriate.

Whilst such waves are not catered for in Heavisides abbreviated equations, I believe that they are admissible in the Maxwell-Dirac quaternion-based system. More here: https://arxiv.org/pdf/1301.4719.pdf

Of course, I could be wrong.

Attach a long copper wire with the plus or minus battery-terminal, and you send a Longitudinal Electro(scalar)Magnetic wave acros the entire copper wire, as transient phenomenon where the entire wire receives a min or plus surface charge. The 'charging' signal travels with the speed of light and it is definitely NOT a TEM wave.

dear Koen, Your statement is incorrect. Yes, in your example, the electromagnetic field will change along the conductor, until at the other end there appears some charge and the potential along the conductor will not be constant. But! This field is not described exclusively by Maxwell's equations. And it does not propagate with the speed of light, but with the speed of perturbation in the electron gas. The electron density, ie, the charge, along the conductor will not be the same, which, indeed, .........

My statement is correct. The entire copper line will receive a surface charge (which requires something different from a mere electron gas disturbance; it is a surface charge current with non-zero divergence) by means of a signal that has velocity almost equal to 'c'. The surface charge on the copper wire is proportional to the line voltage, and the voltage on the copper wire becomes an equipotential with the battery terminal. In many practical situation, the wire surface charge is not that significant, however, it always flows with almost the speed of light before any drift current (which requires a line voltage gradient i.e. a surface charge gradient) can flow through the entire wire cross section. The surface charge signal is not a TEM wave, it has a longitudinal electric field component, and the surface current is such that its divergence is non-zero (otherwise d(rho)/dt would zero, and the wire would not receive a surface charge by means of this signal). Only in case the line voltage becomes high and very dynamical, one can make efficient use of surface charge and surface current on copper wires in order to transfer energy. The old-fashioned single wire Morse Code telegraph system could not have functioned without the existence of longitudinal electro(scalar)magnetic waves, guided by the single wire.

Propagation of the EM field in the single wire is well known task. Its solution is given in Sommerfeld, Electrodynamics. Lectures on theoretical physics, Vol. III, Sec. 22.

According to Sommerfeld, the basic equation for propagation is the differential equation for the longitudinal (along the wire) component. So the speed of propagation is close to the speed of light and not to the velocity of motion disturbation of the electron gas.

Despite I think Sommerfeld made a mistake when he omitted the consideration of the disturbation of the electron gas. But to consider this factor, it is a very hard task. So Sommerfeld's solution is good as the 'zero approximation'.

I apologize in advance, since I only have an elementary understanding of Classical EM field theory, and this is not quite what you asked. At any rate, you might find this interesting, although it most likely belongs in the realm of fringe/pseudo science. It is a document that can be found on the CIA's website regarding scalar EM waves:

https://www.cia.gov/library/readingroom/document/cia-rdp96-00792r000500240001-6

Basically, it shows that there is a solution to the Maxwell/Heaviside equations for a scalar wave, when both E and B fields are zero (i.e. not an EM wave).

The question is whether it is physical, since the Poynting vector would be zero, so there should be no EM power flow.

The document goes on to speculate about the potential source of the energy carried by the proposed scalar wave, and about its nature (e.g. solitons as the result of collective phenomena such as quantum vacuum fluctuations, AKA zero point energy mumbo jumbo).