20:41How Subs Navigate Without GPS for 6 Months. (Its Genius)
Second Order Science opens on the 2005 USS San Francisco grounding, a Los Angeles class boat that hit an uncharted seamount at flank speed, and uses it to ask how a submarine normally knows where it is with no GPS, no light, and no active sonar. The answer is derived rather than described: seawater is a conductor, so the skin depth formula kills a 1.5 GHz GPS signal inside 6.5 mm, forcing dead reckoning built from Newton's second law and two integrations, which turns a 10 to the minus 4 g accelerometer bias into four miles of error in an hour. Maximilian Schuler's 1920s fix was to notice the engineers had assumed a flat Earth and to tune the platform to the period of a pendulum reaching the planet's centre, 84.4 minutes. Ring laser gyroscopes then replaced the spinning flywheels using the Sagnac effect, promptly went blind during slow turns from laser lock in, and had to be cured by shaking the glass block thousands of times a second. It ends on gravity gradiometry, which resets the drift silently against a stored gravity map, and cold atom interferometers aiming at a six month patrol inside a few metres.