Handwerker-Roberto: Primer (Step 4)

Longitudinal "L.M.D." waves from a Tesla coil, quaternionic Maxwell equations, and water antennas

Inventor: Dr. Eng. Roberto Handwerker, DELTA Ingegneria, Milan and Busto Arsizio (Varese), Italy; documents dated 2010-2012 [read]. Verdicts come from arbiter.md, the authoritative ruling; this primer adds no new verdicts. Source tags: [read] = from the project documents or files the Investigator downloaded; [web] = with URL; [memory] = recalled, with confidence; "calc" = recomputed in example.py.

Source gap (repeated from claims.md and arbiter.md). Handwerker's central works are readable: his March 2011 paper "Longitudinal dielectric waves in a Tesla Coil and quaternionic Maxwell's equations" (PDF pages 5-23 of the 208-page bundle TESLA-THEORY-2.pdf), two water-antenna slide decks (August and December 2011), his compilation of Tesla newspaper excerpts (March 2012), and his 2010 deck "Tesla and cold electricity", which the Investigator recovered from the Wayback Machine. Not obtained and not read: the "appendix" and reference [12] his paper cites; his E-book "L.M.D. waves and quaternions" (the likeliest place for a missing derivation); the 41-page Verona deck and the sailboat deck (first pages only); his YouTube videos (channel HorizonDelta, deleted; one 2016 re-upload not watched); his articles in the magazine Heos; and any raw measurement. No document contains a measured value for the water antennas. About 185 of the bundle's 208 pages are other people's Tesla material; they are context, not Handwerker's evidence. Most were read by digest only, but after the QE round the deep reader also read Tesla's testimony and Peterson (pp.24-52), Blazevic et al. and Marincic-Budimir (pp.89-97) and Eckardt (pp.164-172). Of the 29-page 2010 deck, pp.11-22 were read by the Investigator and the physicist, and pp.1-10 and 23-29 were first read by the QE (the arbiter re-extracted pp.24 and 26). Everything said below about his experiments rests on his own reports.

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1. The idea in one paragraph

A Tesla coil is a tall, resonant high-voltage transformer whose top terminal swings to tens or hundreds of kilovolts at radio frequency. Hold a fluorescent tube near one and it lights with no wires; hang a metal plate near it and the plate draws a current and throws sparks. Handwerker set out to show that these effects are not caused by ordinary radio waves, which are "transverse" (their electric field points sideways to the direction of travel), and which he calls T.E.M. (transverse electromagnetic) waves, but by a different kind of wave he called L.M.D. (longitudinal magneto-dielectric), whose electric field points along the direction of travel, like the pressure in a sound wave. He supported this with about eleven simple bench tests around a small coil running at 2.755 MHz, and with a mathematical argument taken from a 2009 paper by Arbab and Satti, who rewrote Maxwell's equations in the older quaternion notation and found an extra "electroscalar" field. He then built "water antennas", glass vessels of water wired to a special high-voltage generator, that lit lamps and ran a small motor in a second vessel, and he quoted Nikola Tesla's 1930s newspaper statements (light is a longitudinal wave in the ether, cosmic rays travel at 50 times the speed of light, relativity is wrong) as authority. The reviewers agree that everything he saw is real and is exactly what the strong electric near field of a Tesla coil does; the open question is whether anything more than that near field is involved, and none of his tests could tell the difference. His work then disappears from the record in 2013.

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2. What you need to know first

2.1 Maxwell's equations and Gauss's law. The four equations (in SI units, vacuum with sources) are: div E = rho/eps0 (Gauss's law); div B = 0; curl E = -dB/dt (Faraday); curl B = mu0 J + mu0 eps0 dE/dt (Ampere-Maxwell). E is the electric field (V/m), B the magnetic field (tesla, T), rho the charge density (C/m^3), J the current density (A/m^2), eps0 = 8.854e-12 F/m the permittivity of free space, mu0 = 4 pi e-7 H/m the permeability of free space, and c = 1/sqrt(mu0 eps0) = 2.998e8 m/s. The point this project turns on is Gauss's law: in empty space (rho = 0) div E = 0, which forbids an electric field that points along its own direction of travel in a plane wave. Read: Griffiths, Introduction to Electrodynamics, chapters 7 (electrodynamics, displacement current) and 9 (electromagnetic waves).

2.2 Displacement current. Maxwell added the term mu0 eps0 dE/dt to Ampere's law so that charge is conserved when a capacitor charges. No electrons cross the gap; the changing electric field in the gap plays the role of the missing current. For a capacitor C driven at voltage amplitude V and frequency f, the current amplitude is I = 2 pi f C V. Griffiths chapter 7.

2.3 Near field and far field of an antenna. A small source oscillating at angular frequency omega = 2 pi f has wavelength lambda = c/f and wavenumber k = 2 pi/lambda. Its field has three parts. Close to the source (distance r much less than lambda/2pi) the reactive near field dominates: the quasi-static field of the charges (falling as 1/r^3 for a dipole, 1/r^2 for a lone charge) and the induction field of the currents (1/r^2). This field stores energy and hands it back each cycle; it can deliver real power only to something nearby that loads it. Far away (r much greater than lambda/2pi) only the radiation field survives, falling as 1/r in amplitude (1/r^2 in intensity), purely transverse, and carrying energy off for good. The crossover is at r = lambda/2pi (the "radian distance"). Crucially, the near field of a dipole has a radial electric component E_r, pointing along the direction from the source: a "longitudinal" field that the textbook already contains. Read: Griffiths chapter 11 (radiation); Jackson, Classical Electrodynamics, chapter 9 (the full dipole fields, near to far); Balanis, Antenna Theory (the "infinitesimal dipole" and its field regions).

2.4 Coupled resonators. Two LC circuits (inductor L, capacitor C) tuned to the same frequency f0 and coupled with coefficient k (0 to 1) no longer ring at f0. They share two normal modes at f(+/-) = f0/sqrt(1 -/+ k), so the ratio of the two frequencies is sqrt((1+k)/(1-k)). A Tesla coil's primary and secondary are such a pair. Read: Taylor, Classical Mechanics, chapter 11 (coupled oscillators and normal modes); the electrical case is the same mathematics.

2.5 Capacitive pickup. A grounded conductor in an electric field E carries surface charge sigma = eps0 E_n, where E_n is the field component normal to its surface (Gauss's law at a conductor). If E oscillates at omega, the charge flows on and off through the ground wire as a current of amplitude i = eps0 omega E_n A for a plate of area A. This is how a Tesla coil lights a tube held in the hand and how modern capacitive wireless power transfer works. Griffiths chapters 2 (conductors) and 7.

2.6 Potentials and gauge. E and B can be written from a scalar potential phi (volts) and a vector potential A (V.s/m): E = -grad phi - dA/dt, B = curl A. Different (phi, A) pairs related by a gauge transformation phi -> phi - d(chi)/dt, A -> A + grad(chi), for any function chi, give the same E and B. A gauge transformation never changes the charges and currents, which are measurable. The Lorenz gauge is the choice div A + (1/c^2) d(phi)/dt = 0, which turns Maxwell's equations into wave equations for phi and A. Griffiths chapter 10.

2.7 Quaternions, from scratch. A quaternion is a number q = a + b i + c j + d k with one real part a and three "imaginary" parts, where i^2 = j^2 = k^2 = ijk = -1. Multiplication is not commutative: ij = k but ji = -k. Hamilton invented them in 1843; Maxwell's 1873 Treatise gave some results in quaternion form beside ordinary components, and in the 1880s Heaviside and Gibbs split the quaternion into a scalar part and a 3-vector part, which is the vector calculus you know (dot product, cross product, div, grad, curl) [memory, high; history checked by the mathematician, web: https://en.wikipedia.org/wiki/History_of_Maxwell%27s_equations]. The key fact for this project: a change of notation cannot create or destroy a solution. A quaternion equation written out in components is a set of ordinary scalar and vector equations; only adding or dropping an equation changes the physics. A short readable source: Kuipers, Quaternions and Rotation Sequences (any chapter introducing quaternion algebra).

2.8 Longitudinal versus transverse waves, and polarisation. In a longitudinal wave (sound in air or water) the disturbance points along the direction of travel; in a transverse wave (a wave on a string, light) it points sideways. Only transverse waves can be polarised: a polarising filter passes light according to Malus's law I = I0 cos^2(theta). Polarisation of light (Malus 1808; Fresnel and Arago, 1810s-1820s) and of Hertz's radio waves (1880s) is the experimental proof that light and radio are transverse [memory, high]. Read: Hecht, Optics, chapter 8 (polarisation).

2.9 Skin depth and lossy media. An electromagnetic wave entering a good conductor of conductivity sigma (S/m) decays over the skin depth delta = 1/sqrt(pi f mu0 sigma). That simple formula needs the loss tangent p = sigma/(omega eps) to be much greater than 1 (conduction current much larger than displacement current), where eps is the material's permittivity. Water is a lossy dielectric: relative permittivity about 80 at low frequency, conductivity from about 0.005 S/m (low-conductivity tap water) to 4 S/m (sea water). At 2.755 MHz, p is 0.41 for 0.005 S/m, 4.1 for 0.05 S/m and 326 for sea water, so only sea water is a good conductor. For the general case the amplitude decays as e^(-alpha x) with alpha = omega sqrt(mu0 eps/2) sqrt(sqrt(1 + p^2) - 1). The 1/e amplitude lengths are then 9.7 m at 0.005 S/m, 1.53 m at 0.05 S/m and 0.15 m for sea water (calc, example.py part 2e). The good-conductor formula would give 4.3 m and 1.36 m for the first two, which is wrong because their loss tangents are not large. So a vessel 10-20 cm across barely weakens a 2.755 MHz field at all. Water attenuates radio-frequency (RF) waves; it does not turn them into a different kind of wave. Griffiths chapter 9 (waves in conductors).

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3. The textbook view

3.1 The field around a Tesla coil

Model the coil's top terminal plus its image in the ground plane as a vertical electric dipole of moment p = q d (charge q, separation d) oscillating at omega. With unit vector r-hat pointing from the source, polar angle theta from the vertical, and a time factor e^(-i omega t), the electric field is [memory, high; Jackson chapter 9]:

E_r = (p cos(theta) / (4 pi eps0)) x 2 (1/r^3 - i k/r^2) e^(ikr)

E_theta = (p sin(theta) / (4 pi eps0)) x (1/r^3 - i k/r^2 - k^2/r) e^(ikr)

The three terms in each bracket are the quasi-static (1/r^3), induction (k/r^2) and radiation (k^2/r) parts. Their relative sizes are 1 : kr : (kr)^2. So at kr = 0.1 the quasi-static field is 100 times the radiated one. Two features matter here:

• The near field has a radial, "longitudinal" electric component, E_r, pointing along the line from source to observer. It falls off as 1/r^3 (and 1/r^2), stores energy, and vanishes in the radiation zone, where only the transverse E_theta (with k^2/r) survives.
• Near the coil the field is mainly electric. A Tesla coil's secondary is high-voltage, low-current, so the electric near field is strong compared with the magnetic one. This is standard Tesla-coil engineering lore [memory, high], and the reviewers all accept it.

3.2 Power without wires

Real power can be taken out of the near field by anything that loads it: a grounded plate (capacitive coupling), a resonant loop (magnetic coupling), a gas tube (the electric field ionises the gas). The best-known modern demonstration is magnetic: an MIT (Massachusetts Institute of Technology) group lit a 60 W bulb more than 2 m from the source by resonant coupling in 2007 (Kurs et al., Science 317, 83) [web: https://news.mit.edu/2007/wireless-0607]. Capacitive wireless power transfer, the electric cousin, is an established engineering field [memory, high]. None of this needs a new wave.

3.3 Why there is no free longitudinal wave in vacuum

Take E = E0 cos(kx - omega t) x-hat (a field pointing along x, travelling along x) with omega = ck. It satisfies the wave equation (1/c^2) d^2E/dt^2 - del^2 E = 0. But div E = E0 k sin(kx - omega t), which is not zero, so in empty space it violates Gauss's law (calc, example.py part 2a). Maxwell drew this conclusion himself in 1865: "the only disturbances which can be propagated are those which are transverse to the direction of propagation" [web: quoted at https://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field]. A longitudinal electric wave needs charge. Where charge is present (a plasma, the electrons in a wire, a conductor), longitudinal charge-density oscillations are standard physics.

3.4 Where Handwerker's starting points are right

Fairness first. Several of his starting points are correct textbook physics, and some are underplayed in popular accounts:

• The field around a Tesla coil is mainly electric ("dielectric", in his and Tesla's word) and does have a strong radial, longitudinal component. Describing a Tesla coil as "just a transformer" leaves out its most striking property, the near electric field around the top load.
• That field does deliver usable power at short range by capacitive coupling, which is a genuinely different mechanism from the magnetic coupling of an ordinary transformer.
• Tesla's own 1891 and 1893 lectures (Handwerker's references [3] and [4]) describe exactly these "dielectric induction" effects, and Elihu Thomson's 1872 observation of strangely electrified objects near an induction coil (his reference [9]) is the same phenomenon [read: p.11, p.23].
• Tesla himself described his "true wireless" as a system that does not radiate. In a question-and-answer excerpt of his testimony bundled in TESLA-THEORY-2 he says "I am not producing radiation with my system; I am suppressing electromagnetic waves ... It is not radiated; it is conserved" [read: p.46], and he chose a large self-inductance to give "a comparatively low frequency, which will reduce the radiation of the electromagnetic waves", passing a current into the earth because that energy is "entirely recoverable, being the energy stored in an elastic system" [read: p.45]. He put the split between earth-current energy and radiated waves at 90/10 or 95/5 (no measurement given) [read: p.46, p.52]. He also judged his own 1890s room demonstrations (lamps lit and a motor run with no wires) to be "a transmission by electromagnetic waves" and "absolutely worthless" for power [read: p.30-31]. Gary Peterson, an advocate of the Tesla system whose article follows in the bundle, adds that the Tesla structure "is specifically designed to have a poor impedance match with free space", stores reactive energy, and may also launch a ground-hugging surface wave "possibly similar to the Zenneck surface wave" [read: p.50-51]. Tesla was a party with a priority stake, and both are single sources for their percentages; their words are quoted here as theirs. Two things follow. Tesla's distinction was radiated versus stored (reactive) or earth-current energy, not transverse versus longitudinal; the word "longitudinal" does not appear on those pages [read: deep read]. And a stored, non-radiating field is exactly the reactive near field of 3.1, which is likely the root of Handwerker's "not T.E.M." intuition. The one standard wave with an electric component along its direction of travel, the Zenneck (transverse-magnetic) surface wave, needs a conducting ground and carries a magnetic field [memory, medium], so it is related to Tesla's earth currents but is not Handwerker's air-borne, non-magnetic L.M.D. wave (arbiter 10.3).
• Steinmetz's remark, which Handwerker quotes, that "the dielectric field is not only confined on the surfaces of the conductors but is also present in the surrounding space" [read: p.18] is the standard field picture.
• Maxwell's equations can be written in quaternion form, and his source, Arbab and Satti (2009), is a real published paper (Progress in Physics vol.2, University of Khartoum and Omdurman Ahlia University) [read: research/files/Arbab-Satti-2009]. Writing fields in terms of scalar functions also has a respectable history: E. T. Whittaker (1904) expressed the electromagnetic field by two scalar potential functions [memory, high], and in quantum mechanics the potentials have measurable effects of their own (the Aharonov-Bohm effect) [memory, high].

3.5 Arbab and Satti's construction, in textbook terms

Arbab and Satti write the potentials as one quaternion and the continuity equation as one quaternion product. Setting the whole product to zero, not only its scalar part, gives ordinary charge conservation (scalar part) plus two extra equations (vector part) [read: Arbab-Satti eqs. 26-27]:

(26) grad(rho) + (1/c^2) dJ/dt = 0, (27) curl J = 0.

From (27), J is the gradient of a scalar: J = grad(Sigma) (their Lambda, Handwerker's Sigma). From (26), rho = -(1/c^2) d(Sigma)/dt. Then Sigma obeys a wave equation at speed c, the "electroscalar wave". Two points the reviewers checked in their paper:

• Its authors say "Maxwell equations are still exact and need no modifications" and call for "urgent experimental work" [read: Arbab-Satti abstract and conclusion]. They do not say Maxwell's vector equations are incomplete.
• Equations (26)-(27) are constraints that ordinary sources break: a static charged ball (J = 0 but grad rho not 0) violates (26), and a current circulating in a loop or coil (curl J not 0) violates (27) [read; reviewers' calc]. So the Sigma field exists only for special charge-current patterns.

What is that pattern? Take Sigma = dPsi/dt with Psi = f(x - ct), any profile moving at c. Then rho = -f'', J = rho c x-hat, and the fields E = -(1/eps0) grad Psi, B = 0 satisfy all four ordinary Maxwell equations (calc, example.py part 2b). So the "electroscalar wave" is a pattern of real charge moving at the speed of light, carrying its own longitudinal electric field with it, and with no magnetic field and therefore zero Poynting vector (E x B/mu0, the energy flux of the field). It is already inside Maxwell theory; it is not a wave in empty space; and ordinary charges, which have mass, cannot move at c. Arbab and Satti suggest vacuum fluctuations might supply such charge, without a calculation [read: Arbab-Satti, after their eq. 36].

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4. Where the inventor departs

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5. Worked example: near field or new wave? Reading Handwerker's own numbers

All numbers below are reproduced by example.py (run from ~/github/altsci with .venv/bin/python review/Handwerker-Roberto/example.py). Inputs come from his paper: f0 = 2.755 MHz [read: p.18]; probe distances 0 to about 2 m [read: page image p.9, Fig.3]; brass plate 120 x 130 mm on a 0-100 mA meter [read: p.10, p.16]; plate readings from about 25-30 mA edge-on to about 110 mA face-on [read: page image p.14, Fig.9]; top bulb 50 mm across [read: page image p.7]. The Fig.3 and Fig.9 values were digitised by eye from the printed plots (deep read; I checked them against my own rendering of p.9 and p.14). They are good to about 5 mA.

Step 1. How far is "near"?

lambda = c/f0 = 2.998e8 / 2.755e6 = 108.8 m. (Handwerker printed 113.64 m, which is c divided by 2.638 MHz: a 4% slip that changes nothing below.) lambda/2pi = 17.3 m, and k = 2 pi/lambda = 0.0577 rad/m.

His measurements stop at about 2 m, so kx is at most 0.115. Every test sat at less than 15% of the radian distance.

Step 2. How big is the stored field compared with the radiated one?

From 3.1, quasi-static : radiation = 1 : (kx)^2, so the ratio of the stored field to the radiated field is 1/(kx)^2:

At his distances the radiated (T.E.M.) field is 75 to 1200 times weaker than the quasi-static field. So when he argues that the radio-wave part "had less possibility of influencing the test" (p.18), he is right. What remains is not an unknown L.M.D. field; it is the quasi-static field of the charged coil, with its radial component, that the textbook predicts.

Step 3. Does the fall-off look like a wave or a near field?

Fit the digitised Fig.3 points (0.47, 1.0 and 1.5 m) to i proportional to x^n (a straight line on a log-log plot):

• triangles (ferrite-coil probe): n = -1.58
• squares (931-A tube probe): n = -1.40
• both together: n = -1.49

Compare: a radiated wave's amplitude falls as 1/x (n = -1); a lone charged sphere's field as 1/x^2 (n = -2); a dipole near field as 1/x^3 (n = -3). The meter current tracks the field amplitude (charge collected per cycle), not the intensity, so Handwerker's dashed "expected quadratic diminishing curve for waves" (Fig.3 caption) mixes up intensity (1/x^2 for a radiated wave) with amplitude (1/x).

Be honest about how soft this number is. Two things the paper does not give move it: adding the near-zero points at 2 m (read as 3 or 5 mA) gives n = -2.1 or -1.85; and the origin of x is not stated (from the axis or from the 200 mm-wide coil form). If x was measured from the coil surface, the distance from the axis is about 0.1 m more, and n becomes -1.68 (or -1.87 for 0.2 m). So the data say "somewhere between -1.4 and -2.1". A fall-off this steep cannot be a radiated wave, which is consistent with Step 2's result that the radiated part is tiny. A value shallower than -2 is not a problem for the near field either: close to an extended charged body (a 210 mm secondary, a probe with a long lead), the field falls more slowly than the far-off 1/x^2 or 1/x^3 limits. The L.M.D. hypothesis states no exponent, so the figure cannot favour it.

Step 4. What field does the plate current imply?

A grounded plate facing a field E oscillating at omega carries current i = eps0 omega E A (section 2.5). Solve for E = i/(eps0 omega A), with A = 0.120 x 0.130 = 0.0156 m^2 and omega = 2 pi x 2.755e6 = 1.731e7 rad/s:

E = i / (8.854e-12 x 1.731e7 x 0.0156) = i / 2.39e-6 (E in V/m, i in A)

• i = 25 mA (edge-on): E = 10.5 kV/m
• i = 100 mA: E = 41.8 kV/m
• i = 110 mA (face-on, above the meter's 100 mA scale): E = 46.0 kV/m

Step 5. Is that field plausible for this coil?

Treat the 50 mm bulb as a sphere of radius a = 0.025 m at potential V. Its field at distance x from its centre is E = V a/x^2, so V = E x^2/a. For the 100 mA reading (41.8 kV/m):

The paper does not give the plate's distance in this test. A 250 mm coil plausibly runs at tens of kilovolts [memory, medium]; a whole charged secondary spreads its charge over a bigger body than the bulb, which lowers these numbers at distances comparable to the coil. So the readings fit a plate within roughly 15-30 cm of the coil, or else the analogue meter did not read true current at 2.75 MHz (meters of this kind with rectifiers are normally rated to tens of kHz [memory, medium]). Either way the readings are within the reach of an ordinary near field and need no new wave; and without the distance, voltage and meter response they cannot be checked further. Note, as the arbiter did, that the 110 mA reading is above the meter's stated scale.

Step 6. The plate-rotation shape (Fig.9).

Rotating the plate about a vertical axis gave a minimum edge-on (psi = 0, plate parallel to the radial direction) and a maximum face-on (psi = +/-90 degrees). The best simple fit to the digitised points is i = 115 - 90 cos(psi) mA (rms residual 3.9 mA), better than |sin psi| (14 mA) or sin^2 psi (8 mA). Minimum edge-on, maximum face-on is the signature of a plate collecting charge from the field component normal to it, which is what Step 4 assumes. Handwerker's own power formula (p.16-17) predicts the opposite (largest at psi = 0), and has a missing resistivity (dR = dx/dS has units of 1/m, not ohms) and a cos^2 that turns into cos. The exact curve shape depends on geometry that nobody, including the reviewers, has modelled. Also, a vertical plate turned about a vertical axis always contains any vertical field, so the test is blind to the vertical "T.E.M." field by construction; "indifferent to Et" is a property of the set-up.

What the example shows. Every number Handwerker printed is what a mainly electric near field of the right size predicts. The calculation does not prove that no L.M.D. wave exists; it shows that his data cannot tell an L.M.D. wave apart from the ordinary near field, because he never says what an L.M.D. wave would do differently. That is why the arbiter rules the strong form "Never properly tested" rather than "Tested, not supported".

Try it yourself. Change f0 in example.py to 0.2 MHz (a large coil) and see where lambda/2pi moves; change the plate area or current; or put in your own readings from a small coil and a neon tube.

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6. Evidence table

Verdicts are the arbiter's (arbiter.md section 9), in the Step 2 scale.

What else is in the record (kept whatever the verdict)

• The 2010 deck (recovered by the Investigator) adds: lamps that stay lit while "short-circuited" by a heavy copper bar (at 2.755 MHz a 0.1 uH bar has reactance about 1.7 ohm, so it is not a zero-impedance short; this is Tesla's old demonstration of inductance at high frequency); a 230 V/100 W halogen lamp lit under water; a flat-spiral coil whose lamp "repels the human hand, but attracts a suspended metal strip"; the rule f1 = (pi/2) f0 for a Tesla coil's two resonances (example.py part 2c: his own 2.755/1.810 = 1.522, which means coupling k = 0.397; pi/2 would need k = 0.423, so the "rule" depends on the coupling); and "MAGNETODIELECTRIC is the dual of ELECTROMAGNETIC" [read: research/files/2011-Handwerker-Tesla-and-cold-electricity, p.11-22].
• More from the 2010 deck and the paper (pp.2 and 5 had not been read before the QE extracted them): he describes himself as "a consulting professional in advanced electrotechnics, thermodynamics & energetics, also technical advisor to italian Judges" (his own description, not independently checked) [read: 2010 deck p.2]; the experiments were "realised by use of low-budget equipment ... partially using scrap parts" [read: p.5]; the deck also shows a vacuum-tube-driven Tesla coil and a receiving plate with a series neon lamp to earth (no data) [read: 2010 deck p.11-22, via sites.md]. The 2011 paper adds that a pocket AM/FM (amplitude- and frequency-modulation) radio picked up a signal out to about the Fig.3 range [read: paper p.15]; 2.755 MHz lies above the AM broadcast band, so what the radio heard is unknown.
• The recurring "pi/2". Three different pi/2 figures meet in this folder: (1) Handwerker's frequency ratio 2.755/1.810 = 1.522, a coupled-resonator number (k = 0.397, above); (2) the rumoured (pi/2)c speed of a longitudinal wave attributed to Dollard (Eric Dollard, an American Tesla experimenter [memory, medium]), which Eckardt reports as "rumors" with "no reliable" experiments [read: TESLA-THEORY-2 p.170]; (3) Blazevic et al.'s "mean surface wave velocity is c.pi/2" [read: p.92]. Item (3) is geometry, not a medium speed: a plane front moving along the earth's axis at c meets the surface at colatitude phi (angle from the pole) at a point moving at v = c/sin(phi), and averaged from pole to equator this is (pi R/2)/(R/c) = c pi/2, where R is the earth's radius. Like a searchlight spot sweeping a wall, the intersection point carries nothing faster than c. The numerical identity (pi/2) x 300,000 km/s = 471,239 km/s is real, and Tesla is often said to have given 471,240 km/s for his earth wave (1904) [memory, medium], but the arbiter's web check did not confirm that figure in his 1904 article, and the Blazevic paper does not state it. So the idea that these pi/2 figures share a source in Tesla lore is an inference, unverified (arbiter 10.6). Physically, Handwerker's frequency ratio has nothing to do with a wave speed.
• Related work bundled in TESLA-THEORY-2 (other authors' context, not Handwerker's evidence; kept whichever way it weighs, arbiter 10.4):
• Horst Eckardt, "What are Scalar Waves?" (2012; a co-author in M. W. Evans's ECE, Einstein-Cartan-Evans, theory programme, a stake) [read: p.164-172]. He defines scalar waves as longitudinal potential waves with E = B = 0 and suggests, without any measurement, transmitters (two antennas half a wavelength apart, or a bifilar flat coil, Tesla's US 512,340) and receivers (capacitor plates half a wavelength apart, a Faraday cage, crystals). In standard theory E = B = 0 means the potentials are pure gauge (2.6) and exert no force. His Faraday-cage receiver is the nearest published design to the grounded-screen test of section 7, though it is drawn ungrounded, with no expected signal or null. He cites Konstantin Meyl's set-up (Journal of Scientific Exploration 15(2), 199-205, 2001) as claiming to use longitudinal components; Meyl 2001 is a second claimed experiment of the same family as Monstein-Wesley, not read or weighed here.
• Marincic and Budimir (Microwave Review, 2001): "Tesla's attempts to formulate a theory of wireless energy transmission proved wrong, but his apparatus built for this purpose was sound"; low attenuation at ELF (extremely low frequency, here 10 Hz to 1 kHz: 0.25 to 20 dB per 1000 km) and an earth standing wave were borne out [read: p.93-97]. The waves they vindicate are ordinary radiated waves in the Earth-ionosphere waveguide.
• Blazevic, Poljak and Cvetkovic (Microwave Review, 2006): a transmission-line model of the Tesla coil, ordinary circuit theory with no longitudinal wave; the authors write that "science considers this theory as fallacious" and prefer coupling to the Schumann (Earth-ionosphere) cavity [read: page images p.90-92].
• Handwerker on Tesla's patents. His preamble says Tesla "issued more than 700 patents, of which ... only about 100 are accessible" [read: TESLA-THEORY-2 p.1]. This is unsourced in the paper. The usual count is about 300 patents in about 26 countries, about 112 of them US [memory, medium; not web-checked]. Reported as his claim with that count beside it.
• The 1932 "neutrons". Tesla's 1932 interview speaks of "neutrons ... carrying so small a charge". Chadwick announced the (uncharged) neutron in February 1932 [memory, high], so Tesla used a brand-new word with a different meaning.
• Coupled-coil interaction (paper p.19): with transmitter and receiver joined by one wire, earth or water, bringing a hand to the transmitter bulb weakens its streamers and strengthens the receiver's, "as true 'conduction' phenomena were involved". In the textbook, loading one of two coupled resonators shifts how energy divides between them; the observation is real and interesting but has no numbers.
• Other Handwerker work known only by name [web: Wayback CDX (the Wayback Machine's capture index) of deltaavalon.com/immagini/]: the E-book "L.M.D. waves and quaternions", "DELTA THOR" earth-pulse and lightning detectors, "DELTA wireless power", the Verona deck (41 pages, on Scribd), the sailboat deck, an earlier 2010 edition of the L.M.D. paper; articles in Heos (2010-2011); products named Xenon G-02 and G-03 Argon. One patent exists in his name, ITVA20010043A1 (2001, an infrared fluid-state monitor), unrelated to L.M.D. [web: https://patents.google.com/patent/ITVA20010043A1/en].
• Others who carried his work: the Tesla Society Switzerland hosted his 2010 deck (captured 28 December 2013, now 404); a Japanese blog summarised the paper as an experimental detection of scalar waves linked to Whittaker; Marco Pizzuti's 2013 popular book Esperimenti scientifici non autorizzati says the Verona demonstration was attended by "several technical experts of the order of engineers" [web: per sites.md]. These are accounts and endorsements; none is a measurement.
• People at the Verona event and leads to missing material. The Heos report names the 15 April 2011 panel: Giovanni Pede (head of ENEA's low-impact vehicles laboratory; ENEA is the Italian national agency for new technologies, energy and sustainable development [memory, high]), Giuseppe Buja (University of Padua, converters, machines and drives), Pietro Menga (president, CEI-CIVES Milan; CEI is the Italian electrotechnical committee and CIVES its electric-vehicle commission [memory, medium]), Bruno Giordano, Marco Tornaghi and Franco Bonfante, moderated by Giovanni Bignami of COSPAR (the Committee on Space Research [memory, high]) [read: research/pages/heos-2011.htm]. Heos co-organised the event (a stake). Mainstream engineers were in the room and are possible witnesses to what was shown; no statement from any of them is on record. The report mentions a Heos "video notizia" (video report), not found or watched. A Facebook commenter says he holds more of Handwerker's materials (sites.md), a lead to the missing E-book, appendix and decks that no one has followed yet. A file "Dedica Gherardo Colombo.jpg" on his old site is recorded in sites.md only (low importance).
• The record goes dark in 2013. His website was normal to 31 March 2013; on 29 May 2013 its homepage showed religious text ("God-believers and in particular Families are under heavy attack of dark evil forces") and every PDF returned 404; by 31 December 2013 the domain was parked; the YouTube channel HorizonDelta was gone by 2021 [web: Wayback, via sites.md]. Google shows "removed in response to a legal request" notices on his name (Lumen notices 75850580 and 95550170, not yet opened). A "Roberto S. Handwerker, Engineer" worked at PT S.p.A. (Società per Azioni, an Italian joint-stock company) 2016-2021 per a LinkedIn snippet; same person likely, not confirmed. The arbiter sets out the competing explanations with equal weight: personal or family crisis, end of a hobby, commercial failure, trade secrecy, legal pressure or suppression (the removal notices exist; such notices are commonly copyright removals; the complainant is unknown), withdrawal after recognising an error, death or incapacity. The evidence favours none of them. A working power device would be expected to leave measurements or a patent, and none exist; that is consistent with an ordinary near-field effect, and equally with any inventor who simply stops. Records that would decide: the Lumen notices, the Italian company register (DELTA Ingegneria, Andromeda di Roberto Handwerker), the Heos articles, and Espacenet for later filings.

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7. What a learner should take away

What is solid. - The near field of a Tesla coil is strong, mainly electric, and has a large radial ("longitudinal") component. Handwerker saw this correctly, and it is underplayed when a Tesla coil is called "just a transformer". - That field delivers real power at short range by capacitive coupling. Neon tubes, sparking plates, a small rectified motor and lamps near water vessels are all within ordinary physics. - His ferrite-core and plate-rotation tests are sensible ideas for telling electric pickup from magnetic pickup. With a grounded screen and a calibrated magnetic probe added, they would become decisive.

What is wrong, and why the mistake is instructive. - Mistaking the near field for "radio waves plus something else". This is the central lesson. Inside lambda/2pi the radiated field is a small remainder (1/75 of the stored field at 2 m here). A strong radial electric field there is not a residue of an unknown wave; it is the field of the charges themselves. Many "longitudinal" or "scalar wave" claims rest on near-field measurements; check kx before anything else. - No monopole radiation: a symmetric source emits nothing, not "only longitudinal". Handwerker argued that the coil's spherical top "cannot emit transverse waves", so it must emit L.M.D. waves. The first half has a true kernel; the conclusion is backwards. By Gauss's law the field outside a spherically symmetric charge depends only on the total charge, which is conserved, so a spherically symmetric source, however its charges move inside it, radiates no wave of any kind; the field near it is just the static-looking Coulomb field Q/(4 pi eps0 r^2), "longitudinal" only in the trivial sense of pointing outward. A real top load is fed through the coil, so with the coil and ground it is a dipole with ordinary near and far fields. When an argument says "not A, therefore B", check that B is not also ruled out. - Notation is not physics. Quaternions, vectors and differential forms describe the same equations. A new wave requires a new or a dropped equation, and you must be able to say which one. - Second-order equations lose information. Differentiating Maxwell's equations into wave equations (1)-(2) throws away Gauss's law; the "extra" longitudinal solutions they admit are artefacts of that loss. Always put the first-order constraints back. - A gauge transformation never touches the sources. If a "transformation" changes rho or J, it is a change of physics, not of description. - Intensity vs amplitude. A meter current tracks field amplitude. A radiated wave's amplitude falls as 1/x; 1/x^2 is its intensity. - A control must be able to see what it is meant to exclude. A compass cannot follow 2.755 MHz. - Units catch errors: "mH" for a capacitance, dR = dx/dS in 1/m, phi in V/m^2. - Mechanical and electromagnetic waves are different things. "Fluids carry only longitudinal sound" says nothing about electric fields in water. - A Newtonian formula outside its range gives faster-than-light speeds. Tesla's 50 c is roughly what v = sqrt(2qV/m) gives for 2e11 V; E = gamma m c^2 never lets v reach c.

What is genuinely open. - Does the E-book "L.M.D. waves and quaternions" or the 41-page Verona deck contain an equation for the L.M.D. field? Without one, the strong claim stays undefined. - Would a grounded wire-mesh screen between coil and probe stop the signal? An ordinary near field must vanish behind it. This is a weekend experiment for anyone with a small Tesla coil, a meter and a mesh (high voltage: take the precautions in Handwerker's own disclaimer). - What exactly was lit in the Verona basin, and where did the return current go (ground, a plate, the audience)? No measurement exists. - Is there any realisable charge distribution, other than charge moving at c, that satisfies both Arbab-Satti constraints (26) and (27)? And what did Ignatiev and Leus (1990) actually measure? - Was Monstein and Wesley's 2002 signal ordinary earth-current T.E.M., as Rebilas argued? The full papers were not read. - Who requested the Google removals, and of what? Why did the record end in 2013? - What was Tesla's basis for 215 GV? - Does the effect change when the spherical top is removed or replaced by a bare coil end or another terminal? That run would test his one stated mechanism directly. - What do Meyl's 2001 experiment (J. Scientific Exploration 15(2)) and Sommerfeld's 1909 surface-wave paper, with Norton's later correction, actually show? Neither was read here. In which case, and when, did Tesla give the testimony quoted in the bundle? - Where did the figure 471,240 km/s, or (pi/2)c, enter Tesla lore? Did Tesla state it in 1904? - What did the Verona panel (Pede, Buja, Menga and others) see, and does the Heos video survive? Does the Facebook commenter's material include the E-book or the 41-page deck?

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8. Check yourself

1. A Tesla coil runs at 500 kHz. What is lambda/2pi? Is a probe at 3 m in the near or far field, and by roughly what factor does the quasi-static field exceed the radiated field there?
2. Handwerker's Fig.3 shows meter current falling roughly as 1/x^1.5 to 1/x^2. He calls the dashed curve "the expected quadratic diminishing curve for waves". What is wrong with that label, and what fall-off would a radiated wave's meter current show?
3. Show that E = E0 cos(kx - omega t) x-hat satisfies the vacuum wave equation but not Gauss's law in vacuum. What extra ingredient would make a longitudinal electric wave legitimate?
4. Why is a compass that stays still near a 2.755 MHz Tesla coil not evidence that the coil produces no magnetic field? Design a control that would detect the RF magnetic field.
5. A 120 x 130 mm grounded plate near a 1 MHz coil draws 20 mA when face-on. Estimate the electric field at the plate. If someone puts a grounded wire-mesh screen between coil and plate and the current falls to near zero, what does that tell you about the L.M.D. hypothesis as Handwerker stated it?

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9. Further reading

Mainstream - D. J. Griffiths, Introduction to Electrodynamics: ch.7 (displacement current), ch.9 (EM waves, waves in conductors), ch.10 (potentials and gauge), ch.11 (dipole radiation). - J. D. Jackson, Classical Electrodynamics: ch.9 (radiating systems; full near-to-far dipole fields). - C. A. Balanis, Antenna Theory: the infinitesimal dipole and the reactive near-field, radiating near-field and far-field regions. - J. R. Taylor, Classical Mechanics: ch.11 (coupled oscillators and normal modes), for the two Tesla-coil resonances. - E. Hecht, Optics: ch.8 (polarisation), for why light is transverse. - J. B. Kuipers, Quaternions and Rotation Sequences: a gentle introduction to quaternion algebra. - A. Kurs et al., "Wireless power transfer via strongly coupled magnetic resonances", Science 317, 83 (2007) [web: https://news.mit.edu/2007/wireless-0607].

The inventor and his sources - R. Handwerker, "Longitudinal dielectric waves in a Tesla Coil and quaternionic Maxwell's equations", DELTA Ingegneria, Milan, March 2011 (revised) [read: TESLA-THEORY-2.pdf p.5-23]. - R. Handwerker, "Tesla and cold electricity" (2010 deck, revised English) [read: research/files/2011-Handwerker-Tesla-and-cold-electricity-revised-English.pdf]. - R. Handwerker, "Water wireless" (25 August 2011) and "H2O wireless" (18 December 2011) decks [read: project folder]. - R. Handwerker (compiler), "Tesla's 'Prophecies' on Cosmic Rays, Relativity and Energy" (March 2012) [read: project folder]. The newspaper originals (Brooklyn Eagle 10 July 1932; NY Times 8 April 1934; NY American and World-Telegram 11 July 1935; Herald Tribune 18 August 1935) have not been checked against archives. - A. I. Arbab and Z. A. Satti, "On the Generalized Maxwell Equations and Their Prediction of Electroscalar Wave", Progress in Physics vol.2 (April 2009) [read: research/files/Arbab-Satti-2009-Progress-in-Physics-electroscalar.pdf]. - N. Tesla, "Experiments with alternate currents of very high frequency" (1891) and "On light and other high frequency phenomena" (1893); US patents 649,621 (1900) and 685,957 (1901); "The True Wireless", Electrical Experimenter, May 1919 (Handwerker's references). - C. Monstein and J. P. Wesley, "Observation of scalar longitudinal electrodynamic waves", Europhysics Letters 59(4), 514-520 (2002); comment by Bray and Britton, EPL 66, 153-154 (2004), reply EPL 66, 155-156; F. Rebilas, EPL 83, 60007 (2008) [web: via arbiter; abstracts only]. - Heos report of the Verona conference, 15 April 2011 [web: Wayback link in section 6]. - Bundled in TESLA-THEORY-2.pdf (context, not Handwerker's evidence): Tesla's testimony excerpt "Nikola Tesla On His Work With Alternating Currents" and G. Peterson's commentary (pp.24-52); D. Marincic and Z. Budimir, Microwave Review, Sept 2001 (pp.93-97); Blazevic, Poljak and Cvetkovic, Microwave Review, Nov 2006 (pp.89-92); H. Eckardt, "What are Scalar Waves?", AIAS/UPITEC (Alpha Institute for Advanced Studies / United Physics for Technology [memory, medium]), 2012 (pp.164-172) [read]. - K. Meyl, Journal of Scientific Exploration 15(2), 199-205 (2001), cited by Eckardt as a claimed longitudinal-wave set-up [not read].

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Corrections to earlier documents

None that change a verdict. QE revision (2026-10-08): (a) my earlier section 2.9 and example.py part 2(e) gave a skin depth of 4.3 m for tap water at 0.005 S/m (1.36 m at 0.05 S/m) from the good-conductor formula, which does not apply there because the loss tangent is 0.41 (4.1); the exact 1/e amplitude lengths are 9.7 m and 1.53 m (sea water stays 0.15 m), and "pure tap water" should have read "low-conductivity tap water". The conclusion (water attenuates; a small vessel barely does) is unchanged. (b) The earlier primer omitted Handwerker's spherical-top argument, Tesla's radioactivity and 1896 priority statements, Tesla's and Peterson's "not radiated; conserved" statements, the related work bundled in TESLA-THEORY-2, the Verona panel and leads, and the fairness point that his 2010 deck has E = -grad phi - dA/dt correct; all are now added with the arbiter's rulings (arbiter section 10). One refinement for the record: the Fig.3 fall-off exponent depends on two things the paper does not give, the reading of the near-zero points at 2 m and the origin of x. My fits give n = -1.4 to -1.6 for the 0.47-1.5 m points alone, -1.85 to -2.1 with a 2 m point, and -1.7 to -1.9 if x is measured from the coil surface (example.py Step 3). The deep read's x^-1.8 / x^-1.5 and the arbiter's "x^-1.5 to x^-1.8" fall inside this range; the conclusion (steeper than a radiated wave, no L.M.D. exponent stated) is unchanged.

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Answers to "Check yourself"

1. lambda = c/f = 3e8/5e5 = 600 m; lambda/2pi = 95.5 m. At 3 m, kx = 3/95.5 = 0.031, deep in the near field. Quasi-static : radiation = 1/(kx)^2, about 1000.
2. The meter current tracks the field amplitude, not the intensity. A radiated wave's intensity falls as 1/x^2, but its amplitude, and so the meter current, falls only as 1/x. A fall-off of 1/x^1.5 to 1/x^2 in amplitude is steeper than any radiated wave and matches the near field.
3. d^2E/dt^2 = -omega^2 E and del^2 E = -k^2 E, so (1/c^2)(-omega^2 E) + k^2 E = 0 when omega = ck. But div E = dE_x/dx = -k E0 sin(kx - omega t), which is not zero, while Gauss's law in vacuum requires div E = 0. A longitudinal electric wave is legitimate when there is charge density to satisfy div E = rho/eps0: a plasma wave, charge oscillations in a conductor, or (Arbab-Satti) charge moving with the wave.
4. A compass needle has far too much inertia to follow a field that reverses 5.5 million times a second; it responds to the average, which is zero. Any RF magnetic field, however strong, leaves it still. Better control: a small shielded loop (a loop of coax with a gap in the shield, which rejects the electric field) connected to an oscilloscope or RF voltmeter, rotated to find the orientation of maximum signal and calibrated against a known coil; compare its reading with the near-field prediction for the coil's current.
5. E = i/(eps0 omega A) = 0.020/(8.854e-12 x 2 pi x 1e6 x 0.0156) = 0.020/(8.68e-7), about 23 kV/m. If a grounded screen kills the current, the signal behaves exactly like an ordinary quasi-static electric field. Handwerker's statement of L.M.D. does not say whether an L.M.D. wave would pass a grounded screen, so strictly the result cannot refute it; this is why the arbiter asks that the L.M.D. claim say beforehand what it predicts. If the claim were sharpened to "L.M.D. is not ordinary E and passes the screen", the near-zero result would count against it.