Tesla, Recalculated

"The wave or wave-train should continue for a certain interval of time, which I have estimated to be not less than one-twelfth or probably 0.08484 of a second and which is taken in passing to and returning from the region diametrically opposite the pole over the earth's surface with a mean velocity of about four hundred and seventy-one thousand two hundred and forty kilometers per second… The lowest frequency would appear to be six per second, in which case there will be but one node, at or near the ground-plate."

United States patent 787,412, filed 16 May 1900, issued 18 April 1905

The simplest possible statement of a planet as a resonator: fit exactly one wave around it. Frequency is speed divided by wavelength, and here the wavelength is the whole way round.

Run it the other way and it is neater still. A wave at 7.83 hertz travelling at light speed is 38,288 kilometres long, against a circumference of 40,030 kilometres. The measured fundamental is very nearly one wave round the world, within four and a half per cent, which is exactly the geometric picture patent 787,412 describes. The arithmetic that gets there is one division.

The textbook treatment models the space between the ground and the lower edge of the ionosphere, the electrically conducting layer of the upper atmosphere, as a hollow shell with perfectly conducting walls. That gives a ladder of allowed rates, one for each pattern the shell will hold.

The measured set used here is the one commonly quoted at 7.83, 14.3, 20.8, 27.3 and 33.8 hertz. Balser and Wagner's own 1960 readings came out slightly different, at 7.8, 14.1 and 20.3 hertz for the first three, and both sets circulate; the conventions differ by a fraction of a hertz and the picture is the same either way.

Now put every figure for the fundamental in one column and the result is the ordering.

The two calculated figures err in opposite directions and by comparable amounts, and the measurement sits between them. The textbook figure overshoots because its upper wall is a perfect conductor and the real ionosphere is lossy and finite, which pulls the observed rate down. Tesla's six per second undershoots. The plainest arithmetic in the set, speed divided by circumference, lands closest of all.

Patent 787,412 states a propagation speed, a transit time, a lowest frequency and a band of observed wavelengths. Those four can be tested against one another without any outside data at all, which is the cleanest kind of check available: it needs nothing but the document.

The document is internally consistent to better than a tenth of a per cent. His velocity is π/2 times the speed of light to five figures, his transit time is one full circumference at that velocity, his six per second is one over twice that transit, and his working band sits just inside the ceiling he gives. Whatever premise produced the factor of π/2, everything downstream of it is exact arithmetic.

One thing the same document does, and it is worth naming rather than smoothing over: it describes two regimes at once. The six per second belongs to the whole-Earth mode. The 25 to 70 kilometre wavelengths belong to the apparatus, and they sit about 1,100 times higher in frequency. Both statements are in the same specification, and they are not the same system.

"A frequency of nine hundred and twenty-five per second would maintain nine hundred and twenty-five stationary waves in a circuit one hundred and eighty-five thousand miles long… each wave would be two hundred miles in length… I would use a secondary of fifty miles in length."

United States patent 649,621, filed 19 February 1900, issued 15 May 1900

This one closes on the last digit of all three numbers, and his velocity sits within 0.7 per cent of light speed. It matters beyond itself, because it is the design rule that carries into everything else he built: the conductor length is a quarter of the wavelength in the system. Every later structure gets measured against it below.

Notebook, 8 October 1899: primary capacity "4 tanks = 4 × 0.03816 = 0.15264 mfd"; primary inductance "Lp = 56,400" centimetres.
Published, June 1900: "The large coil on the right, discharging strongly, is tuned to the fundamental vibration, which is fifty thousand per second."

Colorado Springs Notes 1899 to 1900 · Century Illustrated Magazine, June 1900

An 8.5 per cent close from a notebook capacitance and a notebook inductance, with nothing fitted and nothing adjusted. Landing exactly on 50,000 per second at 56.4 microhenry would need 0.17965 microfarad, which is 4.71 tanks rather than four.

His other published figure, "one hundred thousand alternations per second", is the same machine counted in half-swings. An alternation is one half-cycle, so 100,000 alternations is 50,000 complete cycles. The two published numbers are one frequency in two conventions, and the tank arithmetic supports that reading.

Notebook, 8 October 1899: secondary wire length "803 meters, namely 17 turns each of a diameter of 15 meters"; extra coil "889 meters, namely 149 turns each of diameter of 1.9 meters"; special series coil "307 meters, namely 160 turns each of diam. of 0.61 meter".

Colorado Springs Notes 1899 to 1900

Apply his own quarter-wave rule to those conductor lengths. A wave travels slower along a coiled wire than through free space, and the ratio between the two is the velocity factor. Solve for the velocity factor each length would need to resonate where the tank puts the drive.

A velocity factor of 0.866 is ordinary for a large air-cored helix. So the two independent routes to an operating frequency, one from the capacitance of four jars and the inductance of a loop, the other from the physical length of the coil stack, meet at about 54 kilohertz. That is what a tuned system is supposed to do, and it is the sort of agreement that is difficult to arrange by accident across two pages of a notebook.

"This result is produced by the discharge of an electrical oscillator giving twelve million volts."

Century Illustrated Magazine, June 1900. His 1935 paper gives twenty million volts for the same 1899 machine.

Q is the quality factor: how many swings a resonator takes to lose its stored energy, and equally how many times the voltage rises above what the drive alone would give. Start with skin effect, the way alternating current crowds into the outer skin of a conductor and leaves the middle idle.

A working band of Q of about 300 to 1,500 for a coil of those dimensions. That is the same band as the modern measured ceiling for room-temperature copper: the 2007 MIT self-resonant helix measured 950 plus or minus 50, and industry practice puts a few hundred as typical with about a thousand as the practical top. A coil of the Colorado Springs size sits squarely in it.

Now the twelve million volts, two ways.

The terminal is a 30 inch ball, whose capacitance computes to 42.4 picofarad. The primary is 0.15264 microfarad.

That is the useful distinction, and it falls out of his own numbers without any outside assumption. The twelve million volt figure belongs to a continuously excited resonator, not a single bang. Patent 1,119,732, filed in January 1902 and issued in December 1914, is explicitly about changing that transfer timing. The break rate of 5,000 per second and the supply of 40 kilovolts are both his, from patent 645,576, and they are what close the sum.

The often-repeated "300 kilowatt input" for the station reaches us as a secondary reconstruction rather than a Tesla sentence. His own patent numbers put the machine between 153 and 611 kilowatts across a charging voltage of 20 to 40 kilovolts, and 300 kilowatts falls inside that range at about 28 kilovolts. The reconstruction sits comfortably with the primary-source figures.

"I propose to distribute ten thousand horse-power under a tension of one hundred million volts… a transmitter will emit a wave complex of total maximum activity of ten million horse-power."

Electrical World and Engineer, 5 March 1904

The pairing closes exactly on 75 milliamps. That number is the clearest single signature in the whole set of which regime he was designing in: an impedance six and a half orders of magnitude above free space describes a high-voltage, low-current displacement scheme, not a travelling-wave one. It is not a small detail of the design. It is the design, stated in one ratio, and it is his own arithmetic that states it.

Lamps "lighted to full candle-power by currents induced in a local loop… at a distance of one hundred feet from the primary circuit."

Century Illustrated Magazine, June 1900

Close to a transmitter, energy sits in the surrounding field and comes back; far away, it leaves and does not. The boundary between the two is fixed by wavelength alone, at one wavelength divided by two π.

Two things fall straight out of that table. The 1899 lamps and the 2007 laboratory result are the same regime measured at different scales, and the 1899 one is further inside it. And at six per second the near-field radius exceeds the radius of the Earth, so choosing that frequency puts the whole planet in the zone where energy is stored in the field rather than launched into it. Whatever else the six per second is, it is a choice that keeps everything inside the reactive zone, which is where the lamp demonstration was already working.

Take the 1904 target and let it spread in all directions. Power thins as the surface of a growing sphere, and what a receiver collects is that thinning multiplied by its own catching area.

Radiated outward in all directions, the 1904 power arrives across the Atlantic as 0.13 watts into a resonant element. That is a strong signal and it is a different thing from a power delivery, and the reason it is a strong signal is the low frequency: catching area grows with the square of the wavelength, so one element at 54 kilohertz effectively catches over 3.65 square kilometres. Low frequency buys enormous catching area per element. It is exactly why long-wave telegraphy worked across oceans, and the same arithmetic is what puts power delivery in a different class of problem.

Claim 1: transmitting electrical energy "by conduction, through the earth and the air strata".
1923 testimony on the Wardenclyffe well: "to have a grip on the earth so the whole of this globe can quiver".

United States patent 645,576, filed 2 September 1897 · foreclosure testimony, 1923

Skin depth is the distance an alternating current sinks into a conductor before it drops to about a third of its surface value. It is set by the medium and the frequency, and not by how hard the transmitter is driven.

Every ten decibels is another factor of ten, so 853,000 decibels is 85,300 factors of ten. That single column is the number that sends every practical system to a guided mode along the surface rather than bulk conduction through the crust.

The Wardenclyffe well is the other half of it, and here the arithmetic supports the engineering directly. Spreading resistance is the resistance an electrode meets as current fans out into the ground around it.

The deep well is the right piece of engineering for the mechanism he described, and the arithmetic says why: growing the electrode from one metre to a hundred cuts the spreading resistance a hundredfold. It also says where the loss lives. Ninety per cent of it is within a kilometre of each end, and the path between two distant electrodes contributes almost nothing, which is why earth-return telegraphy worked across continents. The grip on the earth is real. What it grips is a resistive path whose whole cost is at the ends.

Then set his own quarter-wave rule against the largest station of this kind ever built. Project ELF, at Clam Lake in Wisconsin and Republic in Michigan, ran at 76 hertz on about 135 kilometres of ground-dipole line from 1989 until it was dismantled in September 2004.

Wardenclyffe's total vertical structure, 187 feet of tower plus a 120 foot shaft plus 300 feet of driven pipe, comes to 185 metres. That lands at 13.39 per cent of the electrical length the quarter-wave rule asks for at 54 kilohertz. Project ELF landed at 13.69 per cent. The two convergences are within a third of a per cent of each other, which makes Project ELF's measured performance the most directly informative number available for the launching problem: 8 watts radiated from 2.6 megawatts of input, a launch efficiency of three parts per million.

Radiation resistance goes as the square of that fraction, so the scaling is computable. Building the full 986 kilometres would raise the radiated power by a factor of 53, from 8 watts to about 427 watts, still 0.016 per cent efficient. Tesla's own rule and the measurement agree about where the difficulty sits, and they agree in advance of each other by ninety years. Loading coils supply the missing electrical length for tuning; they do not supply the radiating length. The two roles separate cleanly in the arithmetic.

The guided path itself is the cheap part, and this is the piece most easily lost. Measured attenuation in the Earth-ionosphere waveguide at these frequencies is 0.5 to 1 decibel per thousand kilometres, so 1.5 to 3 decibels across 3,000 kilometres, against roughly 9,000 decibels for bulk diffusion through the crust at the same frequency. Split the scheme into propagation and launch and they behave nothing alike.

Propagation across a hemisphere costs ten to twenty decibels, which is affordable for power. Launch is where the entire budget goes, and the measured figure is three parts per million. Launch efficiency is the single quantity the whole scheme turns on, it is a structure-length problem, and the scaling law for it is the one written in his own patent from 1900.

Corona is the soft glow discharge that starts at the surface of a high-voltage conductor once the field there passes a threshold. Frank Peek measured that threshold and wrote it as a formula in which the only variables are the air density and the radius of the surface. Bigger and smoother holds more.

At twelve million volts the Colorado Springs terminal is running about 12.7 times above its own corona threshold, so the machine was in continuous discharge by design rather than sitting near an edge. That is exactly what he described and photographed: a blaze devouring the nitrogen of the atmosphere, sixty to seventy feet across.

The Wardenclyffe cupola tells a different story from the same formula. At 68 feet across its corona ceiling computes to 31.4 megavolts at sea level, comfortably above both his Colorado figures. The tower's terminal was sized for a genuinely quieter high-voltage regime than the one he had been running in, which is a design decision visible in the geometry alone.

Peek's formula run backwards gives the terminal each stated voltage would need in open air at 2,000 metres.

Altitude cuts both ways in the same design, and the two effects are computable separately. Patent 645,576 wants terminals in thin air because thin air conducts sooner: Paschen's law gives 2,234 kilovolts across a one metre gap at sea-level pressure, and 461 kilovolts at 135 torr, a fall of about 4.8 times. The same thinning lowers the corona ceiling on the terminal by the air density factor: the 68 foot cupola's 31.4 megavolts at sea level computes to 10.1 megavolts at 35,000 feet and 6.9 megavolts at 43,300 feet. The conduction he wanted and the corona he did not want move together.

His stated pressures and his stated altitudes describe two different heights, and that is a seam with a number on it. Working the standard atmosphere, 120 millimetres of mercury sits at 43,306 feet and 150 millimetres at 38,663 feet, while 30,000 feet gives 226 millimetres and 35,000 feet gives 179. The gap between what he specified and what he described is 3,663 to 13,306 feet of height, or a pressure ratio of 1.19 to 1.88.

"The flame-like discharge shown in the photograph measures sixty-five feet across… a roaring blaze, devouring the nitrogen of the atmosphere and measuring sixty or seventy feet across."

Century Illustrated Magazine, June 1900. Secondary accounts report 135 feet.

Spark length is the one quantity for this machine that anyone standing there could measure with a tape. Long air sparks break down at a gradient that depends on the shape of the impulse, and three published conventions bracket it.

His published twelve million volts across the 65 foot discharge implies a mean gradient of 6.06 kilovolts per centimetre, a little above the lightning-impulse convention and well inside the range long air sparks show. The three conventions bracket his figure from 2.97 to 19.5 megavolts. The independent measurement supports the order of magnitude he stated, with no adjustment needed to his numbers at all.

Discs "some 9 or 10 in. in diameter, set horizontally, about 1/8 in. apart"; Waterside Station tests at 9,000 revolutions a minute on 18 inch discs with 125 psi inlet.
Patent 1,061,206: "the intervening distance should be the greater, the larger the diameter of the disks".

Technical press, December 1911 · United States patent 1,061,206, filed 17 January 1911

The bladeless turbine works on the thin layer of fluid that clings to a moving surface, the boundary layer. Theodore von Kármán solved the thickness of that layer for a spinning disc, and it depends on the fluid's stickiness and the speed of rotation. Matched design puts the gap at twice the layer thickness, so the layers from both discs just meet in the middle.

For the atmospheric-exhaust condition the Waterside machine actually ran, the eighth-inch spacing is within a factor of two of the matched value. For a machine whose spacing was set by shop practice rather than by this calculation, that is a close result. At the pressurised inlet the same gap is 4.9 times too wide, so the layers do not meet near the inlet and do meet near the exhaust. His patent rule that spacing should scale with disc diameter has the right sign, because larger discs at a fixed rim speed turn more slowly and grow thicker layers.

The other reading of the same operating point is the Reynolds number, which compares the fluid's momentum with its stickiness and tells you whether the flow is smooth or churning.

At the tested operating point the flow in the gap is churning rather than smooth, by roughly two orders of magnitude, so momentum crosses the gap by turbulent mixing rather than by the smooth viscous adhesion the patent describes. That is a statement about the operating point rather than about the principle. The smooth regime the patent describes is reachable, and the figure that reaches it is a gap of 0.04 to 0.2 millimetres at 9,000 rpm, one to two orders finer than 1911 shop practice and well inside what modern micro-machining does routinely. It is the most directly testable design change in the whole turbine set.

His own optimisation rule sharpens it further. Patent 1,061,206 says maximum work comes "when the effective speed of the runner is one-half of that of the fluid". Steam at 8.6 bar expanding to atmosphere carries an available drop of about 357 kilojoules per kilogram, which gives a jet at 845 metres a second. Half of that is 422.5 metres a second, which on an 18 inch disc is 17,649 rpm. The machine was tested at 9,000, which is 51 per cent of his own optimum. At 9,000 rpm the fluid passes the rim at 630 metres a second, above the 523 metres a second sound travels in steam at that temperature; at his optimum the relative speed is 422 metres a second, below it. The difference between the tested point and his stated optimum is not a small tuning question. It moves the stage across the speed of sound.

"An output of 200 horse-power from a single-stage steam turbine with atmospheric exhaust, weighing less than 2 pounds per horse-power… consumption under these conditions of maximum output is 38 pounds of saturated steam per horse-power per hour."
Tesla's own stated efficiency: "80 per cent or even 90 per cent", against existing turbines at "about 62 per cent".

Technical press reporting the Waterside Station tests, December 1911

A steam rate is a complete efficiency measurement in disguise. Turn the pounds of steam into kilograms, divide one horsepower-hour of work by it, and out comes the work extracted per kilogram of steam.

His own reported steam rate supports 43.6 per cent, which sits at the top of the band modern experimental bladeless turbines measure a century later, generally 20 to 40 per cent, and above most of them. That is what his test data carries when it is worked rather than quoted. The widely repeated 97 per cent figure does not appear in these sources at all; the numbers he gave were 80 to 90, and those are a separate question with an arithmetic answer below.

The specific power claim from the same year checks against itself perfectly and against the tested machine not at all, and both of those are informative.

The three numbers in his October 1911 quotation agree with each other to the digit: two and a half pounds at ten horsepower a pound is twenty-five horsepower, and ten against 0.4 is twenty-five times. The 16.44 kilowatts per kilogram they describe sits above a modern large turbofan taken as a whole engine, which places the claim at the boundary of a bare rotor rather than a complete installed plant, while the Waterside figure of 0.822 from the same year is what the complete tested machine delivered. The two numbers are measuring different boundaries around the same object, and they differ by a factor of twenty. Naming which components sit inside the boundary is what would settle it: 200 horsepower at 10 hp/lb is a 9.07 kilogram rotor, and 200 horsepower at 2 lb/hp is a 181 kilogram machine.

The pump numbers, from the same interview, close cleanly and are physically ordinary. Four thousand gallons a minute lifted 360 feet is 271.6 kilowatts of hydraulic work, or 364 horsepower, needing a rim speed of 33 to 46 metres a second, which on an 18 inch runner is 1,370 to 1,938 rpm: a routine speed for a pump. And the 110 horsepower nine-inch machine works out at 6.9 horsepower per gap across the sixteen gaps a two-inch stack holds at eighth-inch spacing, which sits between the 5 and 20 horsepower per disc he stated separately. Three statements agreeing without adjustment.

"He vibrates a weight of approximately 20 pounds at the rate of about 80 per second and with a stroke of about 7/8 inch."
Patents 514,169 and 517,900: the period is "no more dependent upon the pressure applied to drive it, than would be the period of oscillation of a pendulum".

Lecture to the Electrical Congress, Chicago, 25 August 1893 · United States patents 514,169 and 517,900

Three numbers, and a mass on a spring is fully determined by two of them. The spring here is a chamber of trapped air.

His three figures give a consistent resonator, and the isochronism claim closes exactly as stated, because the formula for the period contains no pressure term at all. At fixed stiffness and mass, doubling the drive doubles the swing and leaves the rate alone. The premise underneath the claim is that the spring is linear, and an air spring stiffens as it compresses, so the stiffness rises with amplitude. How wide the pressure range can go before the period shifts is set by that stiffening term, and it is directly measurable on a bench today with a laser displacement sensor and an accelerometer.

The same arithmetic scales, which is what makes it a civil engineering question rather than a curiosity. A steel-framed tower of 3.31 × 10⁸ kilograms swaying with an eight second period has a modal stiffness of 204 meganewtons per metre, one millimetre of sway per 20.8 tonnes of steady side push. At resonance the response is multiplied by Q, and structural Q sits at 10 to 50 rather than the hundreds a bare resonator reaches, because a building loses energy into its joints, its cladding and the ground. Twenty-two newtons applied at the resonant rate gives 1 to 5 micrometres of steady sway. Reaching half a metre would take 2.0 to 10.2 meganewtons. The mechanism is real and the multiplier is exactly Q; the quantity that decides the answer is the damping, and that is a measured property of each structure, not a constant.

Tungsten particle of radius "r = 1/100 c.m." with mass "7686/10¹¹ gram", accelerated to "V = 1,613,000 centimeters or 16,130 meters per second". Terminal potentials given as "6 × 10⁷ volts", with "one hundred million volts" projected.

"The New Art of Projecting Concentrated Non-dispersive Energy Through Natural Media", circa 16 May 1935

First the particle against itself. A tungsten sphere of the stated radius weighs 8.06 × 10^-5 gram, which is 1,049 times his stated mass. Solve the radius back from the mass instead and it comes to 9.84 micrometres, or 1/1016 of a centimetre.

The two figures describe the same tungsten sphere if the radius is a thousandth of a centimetre rather than a hundredth. It resolves to a factor of ten in one exponent, the mass figure carries more significant digits, and once resolved the particle is well specified.

Now the energy, and this is where one object reads two entirely different ways depending on the unit chosen. Both readings are correct arithmetic.

An accelerator concentrates energy into single nucleons. This design spreads it across 4.63 × 10¹⁶ of them. That distinction decides what the arithmetic is about: it is a high-speed impact calculation, not a radiation one, and the two obey different formulas from the first line.

The charge is where the paper's own figures pull against each other, and the gap has a number.

The charge his stated terminal voltage implies sits about five times above what a grain that size holds before its own surface field reaches the scale at which a metal surface is stripped by its own electrical stress. Closing that means a terminal at 309 megavolts rather than 60, a factor of 5.2; or a delivered speed of 7,104 metres a second rather than 16,130; or splitting the charge across a train of grains. The first item on his own list of four inventions is "provisions for imparting to a minute particle an extremely high charge", which is precisely the item that would carry the missing figure. He named the gap before anyone else computed it.

Two more seams sit in the same document and both are arithmetic rather than argument. The stated beam cross-section of one hundred-millionth of a square centimetre is 1.13 micrometres across, while the grain is 19.7 micrometres across: the grain is 17.4 times wider in diameter, and 304 times larger in frontal area, than the beam it is meant to travel in. And the stated trajectory of 64 kilometres sits between two computable bounds. At sea-level air density a grain that size loses its speed over about 0.41 metres. In vacuum it would carry 26,531 kilometres at 45 degrees, two thirds of a lap of the planet. For 64 kilometres to be the drag-limited figure, the air along the path has to be at 7.9 × 10^-6 kilograms per cubic metre, which is the atmosphere at about 85 kilometres up. That is why his first listed invention is "a new form of high vacuum tube open to the atmosphere": the whole scheme turns on what the medium along the path is, and that is the measurement to make.