How Subs Navigate Without GPS for 6 Months. (Its Genius)

At a glance

Second Order Science opens on a wreck and ends on a Bose Einstein condensate. On 8 January 2005 the USS San Francisco, a Los Angeles class nuclear attack submarine, hit an underwater mountain at flank speed, 500 feet down, roughly 350 miles from Guam. Ninety eight crew were injured, one died, and the mountain was not on the chart. The question the video builds on is the one investigators asked: how does a boat with the best navigation gear on earth run into terrain nobody knew was there, and more to the point, how does it normally avoid doing that when it cannot see, cannot receive GPS, and will not turn on active sonar?

The answer is a chain of physics, and the video walks all of it with the equations on screen. Radio dies in seawater because salt water is a conductor, and the skin depth formula puts a 1.5 GHz GPS signal at under a centimetre of penetration. So the boat falls back on dead reckoning, built out of Newton's second law and two integrations in a row, which turns a tiny constant accelerometer bias into an error that grows with the square of time. Fixing that required noticing that the Earth is not flat, which is what Maximilian Schuler figured out in the 1920s and solved with a pendulum as long as the planet's radius, a pendulum whose period is 84.4 minutes. Then the mechanical gyroscopes had to go, replaced by ring laser gyroscopes exploiting the Sagnac effect, which promptly developed their own pathology and had to be cured by physically shaking them thousands of times a second.

What follows is the whole video rebuilt in its order, with the derivations carried out rather than gestured at, plus the arithmetic behind each number the narrator quotes.

The mountain that was not on the chart

On 8 January 2005, the Los Angeles class nuclear powered submarine USS San Francisco was making its way through the Pacific Ocean about 350 miles south of Guam. The boat was more than 500 feet below the surface. The captain had ordered flank speed, and the 6,000 ton vessel was cruising at over 30 knots, around 35 mph.

Then, at around midday, the submarine came to a sudden stop.

One moment it was charging through the water at full speed. The next it had slammed head first into something solid. It had collided with a massive underwater mountain that rose thousands of feet from the ocean floor. But for some reason it did not appear on any of the navigation charts on board.

The force of the collision was devastating. The bow absorbed the impact, crushing the forward sonar dome and buckling internal structures under the enormous stress. Inside the submarine, sailors were hurled into steel walls and equipment. Ninety eight crew members were injured, and one sailor later died from his injuries. The accident became one of the most serious submarine groundings in modern naval history. A billion dollar nuclear submarine had nearly been lost, not to an enemy, but because it struck a mountain no one knew was there.

When investigators began piecing together what had happened, one question stood out above all the others. How could one of the most advanced submarines ever built, equipped with cutting edge navigation systems and sophisticated sensors, crash into an underwater mountain at full speed without anyone realising it was there?

The answer, the narrator argues, reveals a challenge that every submarine faces the moment it goes underwater.

Blindfolded on purpose

Up on the surface, navigation is a solved problem. Your phone, your car, and every commercial airliner on Earth rely on GPS. GPS is a constellation of satellites broadcasting ultra precise clock signals from orbit. When you drive down the street, the antenna inside your phone catches those radio signals, calculates the exact time delay from space, and pinpoints your location to within a few feet.

But the moment a submarine submerges, that entire global satellite network becomes useless, because GPS uses radio waves that are easily absorbed by water.

Light offers no help either. Sunlight is absorbed and scattered by seawater so rapidly that below 600 feet the ocean is pitch black. You cannot look out a window, and you cannot use external cameras.

A submarine commander could order the crew to use active sonar, sending out loud acoustic pings to bounce sound off the terrain ahead. But doing that is the tactical equivalent of turning on a bright strobe light in a dark room full of snipers. An active sonar pulse gives away your exact position, heading and speed to enemy listening posts hundreds of miles away.

So the constraint stack is total. Modern nuclear submarines do not use lights. They do not use cameras. They cannot use GPS, and they almost never turn on active sonar. They travel through the deep ocean completely blindfolded, submerged for months at a time, moving at high speeds through narrow underwater canyons.

And yet, most of the time, they know their exact position on the Earth to within a dozen feet.

The rest of the video is the explanation of how that is possible. It is not a sensor trick. It is the mathematics of inertial sensing, and, as the narrator puts it, engineers turning fundamental constants of the universe into a guidance system that does not need to look at the outside world to know exactly where it is.

Why GPS actually fails underwater

The first question has to be answered properly before the rest makes sense. Why does GPS fail underwater in the first place? It comes down to classical electrodynamics.

GPS satellites broadcast their signals on microwave frequencies, primarily around 1.5 GHz. Air is an electrical insulator. Those radio waves travel through the atmosphere and down to your phone with almost zero resistance.

Seawater is a different medium entirely. It is full of dissolved salt, sodium and chloride ions. That makes seawater an excellent electrical conductor. When an electromagnetic wave enters a conducting medium, the oscillating electric field of the wave forces the free ions in the water to move back and forth, creating electric currents. These microscopic currents act like friction. They drain the wave's energy, converting the radio signal directly into heat.

You can calculate exactly how fast a radio wave dies in water with the skin depth equation:

δ = √( 2 / (ω μ σ) )

Here ω is the angular frequency of the wave, μ is the magnetic permeability of the water, and σ is the electrical conductivity of seawater.

Run the numbers the video quotes. Seawater conductivity is extremely high, around 4 siemens per metre. A GPS signal at 1.5 GHz gives ω = 2π × 1.5 × 10⁹ = 9.42 × 10⁹ rad/s. Permeability is essentially that of free space, μ = 4π × 10⁻⁷ H/m. Plug those in and δ comes out at 6.5 millimetres. Less than 1 cm.

Skin depth is the distance over which the field falls to 1/e, about 37 percent, of its surface value. It is exponential from there, so the collapse is brutal. By the time a GPS radio wave has penetrated just one inch below the ocean surface, the field is down to about 2 percent of what it was, and since power goes as the square of the field, the signal power is down to roughly 0.04 percent. Its energy is almost entirely absorbed.

That single number is why every other technique in this video exists. Radio is not attenuated at depth, it is annihilated within the thickness of a coin.

Dead reckoning, rebuilt from Newton

If a submarine cannot receive signals from the outside world, the crew is forced to use a method of navigation known as dead reckoning. The concept is old and simple. If you know exactly where you started, and you measure every change in speed, every direction, and every turn you make from that moment onward, you can calculate exactly where you are right now.

To do that automatically, aerospace engineers build the mathematics from the ground up, starting with Isaac Newton's second law of motion:

F = m a

If you know the mass of a small sensor inside the submarine, and you measure the mechanical force acting on that mass when the submarine moves, you can calculate the submarine's acceleration.

Acceleration is the rate of change of velocity over time. To get velocity you integrate once:

v(t) = v₀ + ∫₀ᵗ a(τ) dτ

where v₀ is the initial velocity of the submarine when you started measuring.

Velocity is the rate of change of position over time. To get actual position you integrate a second time:

x(t) = x₀ + ∫₀ᵗ v(τ) dτ

where x₀ is the starting coordinate on the map.

This double integration is the foundation of every inertial navigation system, or INS. If a machine can continuously measure acceleration in three dimensions, up, down, left, right, forward, backward, the onboard computer can integrate that raw data twice to track the submarine's position anywhere on Earth.

Simple. But there is a catch, and it is the whole reason this is a hard engineering problem rather than a homework exercise.

Integration drift, the engineer's curse

Consider what happens when your internal sensor has a tiny microscopic measurement error. Say the accelerometer reads a constant tiny measurement bias, call it ε. The sensor thinks you are moving forward just a fraction of a millimetre per second faster than you actually are.

When the computer integrates that constant error once to find velocity, the error grows linearly with time:

v_error(t) = ε t

When the computer integrates that velocity error a second time to find position, the positional error grows quadratically:

x_error(t) = ½ ε t²

This quadratic growth is known as integration drift. For anyone who watched the earlier videos on this channel, the narrator notes, this is another engineer's curse.

Put a number on it, the same number the video uses. Suppose the accelerometer mismeasures acceleration by just one ten thousandth of the acceleration of gravity. That is ε = 10⁻⁴ g = 9.81 × 10⁻⁴ m/s², an error so small it is barely a physical quantity.

That error does not stay small, because of the t² in the equation. Time becomes an enemy.

After one hour of navigation, that tiny bias has accumulated into a position error of ½ × 9.81 × 10⁻⁴ × 3600² = 6,357 metres. Your calculated position has drifted away from your actual position by more than four miles. Note what has happened to the velocity channel at the same moment: v_error = 9.81 × 10⁻⁴ × 3600 = 3.53 m/s, which is nearly 7 knots of imaginary speed the computer now believes in.

After 24 hours of driving blind, the video says, you are off by more than 100 miles. After a week, your computer tells you the submarine is in the middle of a deep trench when you are actually steering directly toward a shallow coastal reef.

Reducing that drift is not just a matter of building a better accelerometer. It requires solving a massive problem with the shape of the Earth itself.

The engineers who assumed the Earth was flat

By the 1920s, engineers building the first mechanical gyroscopes and accelerometers hit a wall. They built precise mechanical sensors, placed them in ships, and watched the systems drift wildly out of control after just a few minutes of travel. They could not figure out why the mathematics was failing.

Then a German engineer named Maximilian Schuler realised the problem. The engineers had built their mathematics assuming the Earth was flat.

Here is the failure in detail. In a standard laboratory on land, an accelerometer measures two things: the acceleration of the vehicle, and the acceleration of gravity g pulling straight down toward the centre of the planet at 9.81 m/s². If your vehicle stays level on a perfectly flat surface, the horizontal accelerometer reads zero, exactly as it should.

But the Earth is a sphere. As a submarine drives forward along the curved surface of the Earth, the direction of down, that invisible vector pointing toward the centre of the planet, continuously changes angle in space. Drive forward without tilting your internal sensors down to account for the Earth's curve, and the horizontal accelerometer is no longer perfectly perpendicular to gravity.

Gravity begins to leak into the horizontal sensor. The accelerometer registers a tiny component of gravity, g sin θ, and mistakes it for forward acceleration. The computer integrates that false acceleration twice, and the navigation position drifts into chaos.

The geometry is unforgiving because θ is set purely by how far you have travelled: θ = d / R. Travel 60 kilometres over the curve and θ = 9.4 milliradians, so the leaked signal is g sin θ = 0.092 m/s². Compare that to the 10⁻⁴ g bias from the last section, which was 9.81 × 10⁻⁴ m/s². The leak is about 94 times larger than the sensor error everyone was worried about. The Earth's curvature was not a correction term. It was the dominant error.

The 84.4 minute pendulum

To fix this, Schuler asked a theoretical question. Is it possible to build a sensor that is completely immune to the acceleration of the vehicle? A sensor that always points toward the exact centre of the Earth, no matter how fast the vehicle moves?

He turned to the mathematics of pendulums. A simple pendulum consists of a mass on a string of length L. The time it takes for that pendulum to swing back and forth, its natural period of oscillation T, is governed by:

T = 2π √( L / g )

If you put a normal pendulum inside a car and slam on the gas pedal, the mass swings backward. The horizontal acceleration disturbs the pendulum.

Schuler realised that if you wanted a pendulum that would never swing backward when the car accelerates, a pendulum that would always point straight down toward the centre of the Earth, then the string length would have to equal the distance from the surface of the Earth to the centre of the Earth. The string length L would have to equal the radius of the Earth R, roughly 6,371 km.

Plug the Earth's radius into the pendulum equation and the mathematics gives an exact time:

T = 2π √( 6,371,000 / 9.81 ) = 5,063 s = 84.4 minutes

This is the Schuler period. And here is the aside the narrator cannot resist: this is the exact same amount of time it takes a satellite to orbit the Earth at sea level. That is not a coincidence, it is the same expression. A circular orbit skimming the surface needs g = v²/R, giving a period of 2π√(R/g), the identical formula. The Schuler pendulum and the impossible sea level satellite are the same 84.4 minutes seen from two directions.

You cannot build a pendulum 4,000 miles long. But Schuler proved you do not have to. If you design a mechanical gimbal system, or program an electronic feedback loop, so that its internal control loop oscillates with a natural period of exactly 84.4 minutes, the system behaves as if it were suspended from the centre of the planet. When the submarine accelerates, the horizontal force pushing on the sensor is perfectly cancelled out by the simulated pendulum dynamics. Gravity can no longer leak into the horizontal measurement.

This discovery, known as Schuler tuning, moved inertial navigation from a theoretical novelty into an operational military technology. It allowed early submarines to stay submerged for hours without their navigation systems blowing up from gravitational errors.

The payoff is worth drawing, because it is the single most important structural fact about an inertial navigator. Schuler tuning does not merely reduce the drift. It changes its shape. The unbounded quadratic curve of Figure 2 becomes a bounded oscillation at the Schuler period, with the same accelerometer bias now producing an error that swings up and comes back down forever instead of running away.

Getting rid of the spinning wheels

Schuler tuning fixed the mathematics. Mechanical systems then produced a new problem of their own: moving parts.

Early inertial navigation systems relied on mechanical gyroscopes, spinning metal flywheels mounted inside complex three axis gimbals. These flywheels spun at thousands of revolutions per minute on mechanical bearings. No bearing is frictionless. Over days of continuous operation, microscopic friction in the gimbal pivots generates heat and creates tiny unwanted torques. Those torques caused the spinning flywheel to precess, to slowly wobble out of perfect alignment.

This mechanical drift meant that every few days the submarine had to come near the surface, extend a periscope or radio antenna, and reset its navigation computer using a star sight or early radio beacons.

That is a doctrinal problem, not just an engineering one. For a nuclear submarine whose only defence is staying hidden deep underwater, having to surface every few days to fix the navigation defeats the entire purpose of being a submarine.

If a crew wanted to stay underwater for six months at a time, engineers had to get rid of the spinning wheels. So they started to measure motion using light.

Ring laser gyroscopes and the Sagnac effect

In the 1970s, aerospace engineers created a device called a ring laser gyroscope. They hollowed out a triangular channel inside a solid block of glass and fired two lasers in opposite directions around the loop.

This relies on a physics principle called the Sagnac effect. Because light travels at a constant speed, if the submarine turns, the laser travelling with the turn takes slightly longer to complete the loop than the laser travelling against it. By measuring that microscopic time difference, the submarine knows exactly how fast it is turning.

The time difference for a loop enclosing area A rotating at rate Ω is:

Δt = 4 A Ω / c²

In a ring laser gyroscope, the glass block acts as a resonant cavity for two counterpropagating laser beams. Instead of measuring a shifted fringe pattern on a screen, the two laser beams inside the glass block combine to form a standing wave of light. When the submarine turns, the physical glass block rotates around that standing wave of light. Optical sensors built into the glass block count the bright and dark nodes of the light wave passing by.

That last sentence is the whole trick, and it is worth pausing on. The standing wave is anchored to inertial space, not to the hardware. The hardware turns underneath it, and all the electronics have to do is count nodes going past. Because the wavelength of light is extremely small, less than one micrometre, a ring laser gyroscope detects rotation rates as slow as one revolution every few years. In degrees per hour, that is a bias on the order of a hundredth of a degree per hour, which is roughly what navigation grade specifications call for.

There were no bearings to wear out, no flywheels to balance, and no mechanical friction.

Laser lock in, and the fix that should not work

But as engineers built the first production laser gyroscopes, they hit a fatal optical flaw.

When the submarine turned very slowly, the rotation rate dropped near zero. At these low speeds, a tiny fraction of the light from one laser beam scattered off microscopic imperfections in the mirrors and mixed with the opposite laser beam. This back scattering caused the two laser beams to lock on to the exact same frequency. The standing wave collapsed.

This phenomenon, known as laser lock in, created a massive dead zone. When the submarine made slow, gentle turns, the laser gyroscope went completely blind. It registered zero rotation when the ship was actually turning.

Which is the worst possible failure mode, because a slow gentle turn is what a submarine does most of the time.

To solve this, engineers came up with a bizarre mechanical solution: mechanical dithering. They mounted the entire glass laser block on a central piezoelectric spring. A motor physically shakes the entire laser block back and forth, vibrating it thousands of times a second at high frequency. By constantly shaking the gyroscope, they force the system out of the lock in dead zone. The computer then subtracts the known vibration of the motor, leaving a clean, precise measurement of the submarine's actual rotation.

The narrator's summary of the irony is the best line in the video. They built a system where lasers measure the behaviour of light, while a crude motor shakes the sensor violently to keep the light from breaking itself.

By combining three ring laser gyroscopes with three quartz accelerometers on a Schuler tuned platform, engineers created the modern electromechanical navigator. For the first time, a nuclear submarine could dive underwater in Connecticut, travel thousands of miles under the Arctic ice cap, and emerge weeks later in the Pacific Ocean with its navigation computer off by less than a few hundred yards.

Everybody uses this

The science mapped out here, RF skin depth, double integration, Schuler tuning and the Sagnac effect, are not secret formulas any more. They represent universal laws of nature. Every major naval power on Earth uses them.

You find these same fundamental principles applied in the navigation suites of American Virginia class attack submarines, Russian Borei class ballistic missile ships, and British Vanguard class deterrent vessels. All of them rely on ring laser gyroscopes and quartz accelerometers to navigate in the dark.

Gravity gradiometry, the silent reset

But no matter how precise your optics are, no inertial system is perfect. Over weeks and months at sea, microscopic errors still accumulate. Integration drift never sleeps. Eventually a submarine must update its position.

And there is the tactical trap. Surfacing to catch a GPS signal exposes the ship's sail to radar satellites and maritime patrol aircraft. Every position fix is a moment of vulnerability, which means the navigation problem and the survival problem are the same problem.

So modern navies developed a way to reset their position without surfacing: gravity gradiometry.

The Earth's gravitational field is not uniform. Dense underwater structures like basalt mountains, oceanic trenches and heavy iron deposits create tiny local variations in the strength and direction of gravity beneath the hull. As the submarine glides through the water, gradiometer sensors continuously measure the microscopic changes in the gravitational field beneath the hull.

The onboard computer compares these live gravity measurements against a high resolution map of the seafloor's gravitational field stored in its memory bank. If the computer measures a specific gravitational bump, it matches that bump to the digital map, instantly resetting the double integration drift without emitting a sound wave or exposing an antenna above the water.

It is the same idea as terrain contour matching in a cruise missile, except the terrain being matched is the shape of the gravity field rather than the shape of the ground, and it works through a mile of opaque water.

Cold atoms, and the end of the fix

And the technology continues to evolve. Engineers are currently moving beyond laser gyroscopes toward the next frontier of guidance physics: quantum cold atom sensors.

Inside a quantum navigation system, lasers freeze a cloud of rubidium 87 atoms down to a fraction of a degree above absolute zero. This extreme cold forms a state of matter known as a Bose Einstein condensate. At these temperatures the atoms stop behaving like solid billiard balls and begin behaving like quantum matter waves.

By splitting and recombining these atomic waves using laser pulses, physicists create atom interferometers. The architecture is the same as the ring laser gyroscope, one wave sent two ways and recombined, but the wave is made of matter instead of light.

That substitution is the entire point. Because atoms possess mass, they are millions of times more sensitive to acceleration and gravity than weightless photons of light. A quantum inertial navigation system will allow submarines to travel underwater for an entire six month deployment without drifting by more than a few metres, rendering external navigation signals permanently obsolete.

The video closes on the arc. The story of underwater navigation evolved from sailors guessing their location with magnetic compasses into a high stakes arena of computational optics and quantum physics.

The chronology in one column

• 1920s Mechanical gyroscopes and accelerometers go to sea and drift wildly within minutes. Nobody can find the bug, because the bug is the assumption that the Earth is flat.
• 1920s Maximilian Schuler identifies the gravity leak and derives the fix: tune the platform's control loop to the period of a pendulum reaching the centre of the Earth, 84.4 minutes. Inertial navigation becomes an operational military technology.
• 1950s to 60s Gimballed mechanical systems serve, but bearing friction forces a star sight every few days, and surfacing is exactly what a ballistic missile submarine must not do.
• 1970s The ring laser gyroscope replaces the flywheel with the Sagnac effect. No bearings, no wear. Then laser lock in blinds it during slow turns, and mechanical dithering fixes it by shaking the block thousands of times a second.
• Modern Three ring laser gyroscopes plus three quartz accelerometers on a Schuler tuned platform. Connecticut to the Pacific under the ice cap, off by a few hundred yards. Virginia, Borei and Vanguard class boats all run on the same physics.
• 2005 USS San Francisco strikes an uncharted seamount at flank speed, a reminder that a perfect position on a wrong chart is still a collision.
• Now Gravity gradiometry resets the drift silently against a stored gravity map, and cold atom interferometers aim at a six month patrol inside a few metres.

Where it stands

The physics in this video is standard and correct. Skin depth, double integration, Schuler tuning and the Sagnac effect are textbook, and the derivations on screen are the ones you would find in an inertial navigation course. Three notes worth adding for anyone checking the arithmetic.

The 24 hour drift number does not follow from the quadratic law the video just derived. If ½εt² gives four miles at one hour, then 24 hours is 576 times that, which is about 2,275 miles, not "more than 100 miles". Both statements are in the video within about ninety seconds of each other. The larger number is what the stated equation actually produces.

Real inertial navigators do not drift quadratically, which is the whole reason Schuler tuning exists. Once the platform is tuned, a constant accelerometer bias produces the bounded oscillation of Figure 4, not the runaway curve of Figure 2. Published drift figures for naval inertial systems are quoted in nautical miles per day, growing roughly linearly, and the dominant contribution is gyroscope bias rather than accelerometer bias. The video presents the quadratic case and the Schuler fix in the right order, but does not come back to say that the fix retires the quadratic curve. That is the single most useful thing to carry away from the derivation.

The San Francisco grounding was a charting and voyage planning failure more than an inertial navigation failure. The public record of the Navy's court of inquiry found that the chart in use did not show the seamount, while other materials available on board did indicate discolored water in that area, a classic hazard signature, and that those materials were not consulted when the track was planned. The boat's own sense of where it was appears to have been broadly right. What was wrong was the map. That does not weaken the video's framing, it sharpens it: an inertial navigator tells you where you are, never what is in front of you, and the two failures look identical from inside the hull.

One smaller point of scale. The claim that atoms are "millions of times more sensitive" than photons is conservative for the underlying physics. The Sagnac phase for a matter wave beats the optical case by roughly mc²/ħω, about ten orders of magnitude for rubidium 87 against an optical photon. Practical devices give most of that back to short interrogation times and small enclosed areas, which is why cold atom navigators are still a frontier rather than a fleet fit.

Key takeaways

• Seawater does not attenuate GPS, it erases it. At 1.5 GHz with σ = 4 S/m the skin depth is 6.5 mm, so an inch of water leaves 2 percent of the field and 0.04 percent of the power. Everything else follows from that one number.
• A submarine is blind by choice as well as by physics. No lights, no cameras, no GPS, and active sonar is a strobe light in a room full of snipers. The stealth requirement is what makes the navigation problem hard.
• Dead reckoning is Newton's second law integrated twice, and that second integral is where the danger lives. A constant bias ε gives velocity error εt and position error ½εt².
• A 10⁻⁴ g bias is four miles of error in one hour. Time is the enemy, not the sensor, because the error grows with the square of it.
• The Earth's curvature was a bigger error than the sensors. Carry an unlevelled accelerometer 60 km along the curve and gravity leaks in at 0.092 m/s², about 94 times a 10⁻⁴ g bias.
• Schuler's answer was a pendulum as long as the Earth's radius, period 2π√(R/g) = 5,063 s = 84.4 minutes. You simulate it in the control loop rather than building it, and the platform then behaves as if hung from the planet's centre. The same expression gives the orbital period of a satellite skimming sea level.
• Ring laser gyroscopes traded friction for optics, sensing rotation through the Sagnac effect down to about one revolution every few years, with no bearings to wear out.
• Laser lock in blinded them exactly where it mattered, during slow turns, and the fix was to bolt the glass block to a piezoelectric shaker and subtract the known vibration.
• Every position fix is a tactical exposure, which is why gravity gradiometry matters: match the local gravity anomaly against a stored map and reset the drift without a ping or an antenna.
• Cold atom interferometry is the endgame, aiming at a six month patrol with a few metres of drift and no external fix at all.

Chapters

• 0:00 Intro
• 3:53 Why GPS Fails Underwater
• 8:45 Dead Reckoning & Integration Drift
• 14:04 The 84.4-Minute Schuler Pendulum
• 17:34 Outro

Notable quotes

"One moment, it was charging through the water at full speed. The next, it had slammed head on into something solid." Second Order Science, 0:28

"A billion dollar nuclear submarine had nearly been lost, not to an enemy, but because it struck a mountain no one knew was there." Second Order Science, 1:15

"But doing that is the tactical equivalent of turning on a bright strobe light in a dark room full of snipers." Second Order Science, 2:49, on active sonar

"They travel through the deep ocean completely blindfolded, submerged for months at a time, moving at high speeds through narrow underwater canyons. And yet, most of the time, they know their exact position on the Earth to within a dozen feet." Second Order Science, 3:15

"By the time a GPS radio wave penetrates just one inch below the ocean surface, its energy is almost entirely absorbed." Second Order Science, 8:15

"This double integration is the foundation of every inertial navigation system." Second Order Science, 10:12

"Time becomes an enemy." Second Order Science, 11:37, on the t² in the drift equation

"The engineers had built their math assuming the Earth was flat." Second Order Science, 12:46, on the 1920s failures Schuler diagnosed

"If you plug the Earth's radius into the pendulum equation, the math gives an exact time, 5,063 seconds, or 84.4 minutes. This is the Schuler period." Second Order Science, 15:54

"This discovery, known as Schuler tuning, moved inertial navigation from a theoretical novelty into an operational military technology." Second Order Science, 17:11

"They built a system where lasers measure the behavior of light, while a crude motor shakes the sensor violently to keep the light from breaking itself." Second Order Science, 19:12, on mechanical dithering

"For the first time, a nuclear submarine could dive underwater in Connecticut, travel thousands of miles under the Arctic ice cap, and emerge weeks later in the Pacific Ocean with its navigation computer off by less than a few hundred yards." Second Order Science, 19:20

"If the computer measures a specific gravitational bump, it matches that bump to the digital map, instantly resetting the double integration drift without emitting a soundwave or exposing an antenna above the water." Second Order Science, 20:08, on gravity gradiometry

"The story of underwater navigation evolved from sailors guessing their location with magnetic compasses into a high stakes arena of computational optics and quantum physics." Second Order Science, 20:36

Resources mentioned

The video and the channel

• How Subs Navigate Without GPS for 6 Months. (Its Genius), the source video
• Second Order Science, the channel

The incident

• USS San Francisco (SSN 711), the boat and the 2005 grounding
• Los Angeles class submarine
• Guam, roughly 350 miles north of the collision site

The physics the video derives

• Global Positioning System, the official US government GPS reference
• Skin effect and skin depth, the δ = √(2/ωμσ) result
• Electrical conductivity of seawater
• Dead reckoning
• Newton's laws of motion
• Inertial navigation system
• Accelerometer
• Schuler tuning and the 84.4 minute Schuler period
• Pendulum period, T = 2π√(L/g)
• Gyroscope and precession
• Ring laser gyroscope
• Sagnac effect
• Injection locking, the general phenomenon behind laser lock in
• Piezoelectricity, the dither actuator
• Gravity gradiometry
• Gravity anomaly
• Bose Einstein condensate
• Rubidium 87
• Atom interferometer
• Laser cooling

The boats named

• Virginia class attack submarine, United States
• Borei class ballistic missile submarine, Russia
• Vanguard class ballistic missile submarine, United Kingdom

Music credited in the description

• Arthur Vyncke, "Refusing The Ultimatum", used under Creative Commons BY SA 3.0
• BreakingCopyright, the music library the track came from