The LWave is the traveling (and/or stationary) longitudinal counterpart to the traveling electromagnetic wave (TEM). Using the terminology from Maxwell's original treatises, it can be written as a longitudinal wave in the electromagnetic momentum where the electromagnertic momentum is curl-free (or nearly so). Langmuir's electrostatic plasma wave is one concrete example of an LWave. This post is intended to provide a brief overview of the LWave's history.
James Clerk Maxwell
Maxwell invented the concept of longitudinal waves, and dismissed their existence. Maxwell imagined a conceptual fluid, later called the ether, filling all space [SIMP97]. He imagined transverse waves in the fluid, and then recast the fluid flow equations into the field equations that eventually became Maxwell's laws. Real fluids carry both transverse and longitudinal (compression) waves. Maxwell conceived the idea that his conceptual fluid could carry longitudinal waves. In this sense, he invented the LWave. Unfortunately, the equipment of Maxwell's time was not sensitive enough to detect the extremely small mass of the electron. To within the sensitivity of the equipment, charge carriers appeared to be massless and appeared to move without any inertial lag. A longitudinal wave solution can not exist in the electromagnetic field without some inertial lag on the part of the charge carriers. As a result, Maxwell removed longitudinal wave solutions from his final set of equations. This led to a rather strange set of characteristics for the ether. Maxwell created a fluid that was completely incompressible, hence no longitudinal waves, but yet could carry transverse waves. Although this kind of fluid is possible in theory, it is hard to fathom such an entity really existing.
It seems reasonable to believe that if Maxwell had known the mass and charge of the electron then he probably would have published a different set of equations based on a compressible fluid with longitudinal wave solutions.
Nikola Tesla was the first to claim that he had observed traveling longitudinal waves. Tesla's writings leave one with the impression that Tesla was familiar with Maxwell's conceptual fluid, and that Tesla believed he had discovered the longitudinal waves that Maxwell had dismissed. The evidence suggests that Tesla would have been one of the first people to observe effects that were not known to Maxwell. Tesla was a pioneer in the investigation of microwave frequencies, and in the investigation of plasmas. He invented fluorescent lighting.
Tesla, based on his own experimental results, was a critic of the ether's supposed incompressibility. In [NEWY29], he stated, "When Dr. Heinrich Hertz undertook his experiments from 1887 to 1889 his object was to demonstrate a theory postulating a medium filling all space, called the ether, which was structureless, of inconceivable tenuity and yet solid and possessed of rigidity incomparably greater than that of the hardest steel. He obtained certain results and the whole world acclaimed them as an experimental verification of that cherished theory. But in reality what he observed tended to prove just its fallacy." Tesla believed the ether to be compressible like a gas, and thus capable of carrying compression waves.
He also stated about concerning the excitation of gas in Geissler tubes to produce light, “Long transverse [electromagnetic] waves cannot, apparently, produce such effects, since excessively small electromagnetic disturbances may pass readily through miles of air. Such dark waves, unless they are the length of true light-waves, cannot, it would seem, excite luminous radiation in a Geissler tube, and the luminous effects which are producible by induction in a tube devoid of electrodes, I am inclined to consider as being of an electrostatic nature. To produce such effects, straight electrostatic thrusts are required...” [SHEN81]. In other words transverse electromagnetic waves can not explain how Geissler tubes are excited to produce light, and instead “straight electrostatic thrusts,” i.e. electrostatic thrusts running longitudinal to the axis of the tube, are required.
Moreover, Tesla claimed that his equipment could transmit waves that were non-Hertzian. In [TESL04] he described his Magnifying Transmitter as, “... essentially, a circuit of very high self-induction and small resistance which in its arrangement, mode of excitation and action, may be said to be the diametrical opposite of a transmitting circuit typical of telegraphy by Hertzian or electromagnetic radiations.” In other words, Tesla claimed he was using a mode of excitation that was not electromagnetic. Tesla often used the phrases "Hertzian radiations" or "Hertzian waves" to refer to TEM waves.
In respect to his remotely controlled automaton, Tesla stated in [TESL00a], "... no thoroughly satisfactory control of the automaton could be effected by light, radiant heat, Hertzian radiations, or by rays in general... These requirements made it imperative to use... waves or disturbances which propagate in all directions through space, like sound, or which follow a path of least resistance..." Once again, Tesla draw a distinction between his waves and Hertzian waves.
In another source he referred to his waves as compressions and rarefactions of the ether. Tesla's longitudinal waves, also called scalar waves, have been met with a combination of derision by some and fanciful claims by others. Nevertheless, Tesla remains one of the most prominent proponents of longitudinal wave propagation.
Before Einstein published the Special Theory of Relativity the ether was considered to be an entity that flowed in a physical sense through a classical coordinate system. In this coordinate system, it was valid to add or subtract a velocity to the speed of light. Eventually, Michelson and Morley devised an experiment using this idea of velocity addition to determine how fast the ether was moving. Their experiment always returned one answer for the ether velocity: zero. Einstein fixed this problem in the Special Theory of Relativity by introducing relativistic coordinate systems. Each inertial reference frame has its own coordinate system, and velocities do not add. The speed of light is the maximum velocity in all reference frames. As a consequence of relativistic coordinates, each inertial reference frame would see its own ether. Each inertial reference frame also perceives its own version of time, its own version of relative velocities, and its own version of momenta. In the case of the ether, however, the presence of transforms were considered to be dismissive of its existence. That is to say, science didn't dismiss the ideas of time, velocity and momentum based on relativity, but it did dismiss the existence of the ether. This appears to have adversely affected the acceptance of LWaves. If one doesn't believe that the ether exists, then why would one believe that it has compression waves?
During his employment at General Electric, Langmuir worked to improve the same fluorescent lighting that Tesla had invented. Like Tesla, he investigated plasma physics. By this time the mass and charge of the electron had been established, and Langmuir realized that inserting these constants, plus some Newtonian mechanics, into Maxwell's equations provided an explanation for the experimental results that he was observing. It wasn't a reformulation of Maxwell's ether from first principles, but it was a band-aid. Langmuir also had a success in getting his results published and accepted that eluded Tesla. Langmuir's publications were a step forward in that they added a longitudinal wave mode traveling at the thermal velocity (i.e. the speed of sound) of a medium. The Langmuir wave was the first LWave to be accepted by the scientific community. Maxwell and Tesla probably would have called these changes fundamental. Instead, they were called plasma physics.
Yakir Aharonov and David Bohm
One of the most important variables in Maxwell's treatises was the electromagnetic momentum vector A (which was later renamed the magnetic vector potential by other authors). Maxwell wrote many of the equations in the treatises in terms of the electromagnetic momentum rather than the force fields [SIMP97]. From a reading of large excerpts of Maxwell's works, it seems obvious that Maxwell considered the electromagnetic momentum to be an actual physical quantity. In the post-Heavyside gauge version of Maxwell's equations, the electromagnetic momentum is a non-physical quantity that can be arbitrarily renormalized. The elecromagnetic momentum also appears (under its new name) when magnetic fields are integrated into Schrodinger's equation.
Aharonov and Bohm published a prediction that altering the elecromagnetic momentum, even in cases where no magnetic field is present, would alter the phase of a Schrodinger electron wave, and that this phase difference could be detected using a quantum interference device. In this way, they suggest that the electromagnetic momentum can be uniquely determined. In experiments, this was found to be true [IMRY89]. This had the side-effect of establishing the electromagnetic momentum as a physical property of nature [FEYN66], and making the gauge renormalization of Maxwell's equations invalid in the context of quantum mechanics. Changes in the electromagnetic momentum can apparently be modulated and transmitted. Several patents for such transmission were granted to Raymond Gelinas [GELI84a], [GELI84b], [GELI84c], [GELI84d], [GELI85], [GELI86a], [GELI86b]. Any longitudinal wave transmitted in such a fashion would be an LWave.
All of this tends to suggest that LWaves may be more properly written was longitudinal waves in a curl-free electromagnetic momentum rather than as longitudinal electric field waves in a curl-free electromagnetic momentum. The momentum waves can completely represent the electric field waves, but the electric field waves can't necessarily represent the momentum waves in quantum mechanical cases where the electric field is zero.
Lorentz proposed a gauge, called the Lorentz gauge, for the renormalization of the post-Heavyside version of Maxwell's Equations. Lorentz's gauge is relativistically invariant, which seems to be the primary reason for adopting it. The Lorentz gauge has longitudinal wave solutions in the electomagnetic momentum that are in some respects similar to Langmuir waves. If an appropriate longitudinal current travels in parallel with the longitudinal electromagnetic momentum, then the electric field in the solution becomes non-zero and the resulting wave has a similar electric field solution to that of a Langmuir wave. However, there is a big difference in wave velocities between Langmuir LWave solutions and Lorentz LWave solutions. Langmuir waves travel much closer to the speed of sound than they do to the speed of light. This is passably reasonable. One would expect waves of longitudinally moving electrons to travel much slower than the speed of light, unless accelerated to very large energies, due to relativistic constraints. The Lorentz LWave solutions, which do not incorporate the mass of the electron, produce wave velocities that seem impossibly large without an enormous input of energy.
Nevertheless, the Lorentz gauge is useful conceptually in that it attempts to address relativistic invariance for LWaves. It also shows that more work need to be done in order to completely define a wave equation for LWaves. A more complete reformulation of Maxwell's equations for LWaves would have to address both thermal wave velocities and relativistic invariance in a consistent fashion.
WORKS CITED (Informal)
FEYN66 Feynman, Richard P., Leighton, Robert B., and Sands, Matthew, “The Feynman Lectures on Physics”, Vol. II, Addison-Wesley, Menlo Park, CA, 1966
GELI84a Gelinas, Raymond C., “Apparatus and Method for Demodulation of a Modulated Curl-Free Magnetic Vector Potential”, US patent 4,429,280.
GELI84b Gelinas, Raymond C., “Apparatus and Method for Modulation of a Curl-Free Magnetic Vector Potential Field”, US patent 4,429,288.
GELI84c Gelinas, Raymond C., “Apparatus and Method for Transfer of Information by means of a Curl-Free Magnetic Vector Potential Field”, US patent 4,432,098.
GELI84d Gelinas, Raymond C., “Apparatus and Method for Determination of a Receiving Device relative to a Transmitting Device Utilizing a Curl-Free Magnetic Vector Potential Field”, US patent 4,447,779.
GELI85 Gelinas, Raymond C., “Josephson Junction Interferometer Device for Detection of Curl-Free Magnetic Vector Potential Fields”, US patent 4,491,795.
GELI86a Gelinas, Raymond C., “Apparatus and Method for Distance Determination between a Receiving Device and a Transmitting Device utilizing a Curl-Free Magnetic Vector Potential Field”, US patent 4,605,897.
GELI86b Gelinas, Raymond C., “Curl-Free Vector Potential Effects in a Simply Connected Space”, Paper, Proceedings of the 1986 International Tesla Symposium, International Tesla Society, 1986.
IMRY89 Imry, Yoseph, and Webb, Richard, "Quantum Interference and the Aharonov-Bohm Effect", Scientific American, April 1989, pp. 56-62.
NEWY29 "Nikola Tesla tells of New Radio Theories", article, New York Herald Tribune, September 22, 1929. Recopied in http://www.tfcbooks.com/tesla/1929-09-22.htm
SHEN81 Shennan, A., “Nikola Tesla-- Savant Genius of the Twentieth Century”, Historian Publishers, Tarzana, California, 1981. [Researchers beware: there are multiple versions of this text with the same author, title, and publication date. The version of the text that I reference here can be uniquely identified by the fact that it has a diagram on page number six titled “Forms of Waves Possible with Tesla Transmitter” and showing a cross-section of the Earth.]
SIMP97 Simpson, Thomas K., “Maxwell on the Electromagnetic Field”, Rutgers University Press, New Brunswick, New Jersey, 1997.
TESL00a Tesla, Nikola, “The Problem of Increasing Human Energy”, article, The Century Illustrated Monthly Magazine, June, 1900.
TESL04 Tesla, Nikola, “The Transmission of Electric Energy Without Wires”, article, The Electrical World and Engineer, March 5, 1904.