It's not a stretch to imagine that someone noticed how the length of an echo relates to distance. The acoustical engineering of the bell seems to require some math that late-medieval people didn't have – but then, you could say that about bells in general, and they managed to figure it out by experimentation, without understanding any of the math. The materials are no problem – brass works OK in seawater, and the membrane for the pickup could be mica or isinglass.
The mechanics aren't beyond medieval engineering. So you have a chart recorder that will record the initial ping as a large mark, and its echo as a smaller mark, with the distance between them measuring the time taken and therefore the distance to the sea floor. The bell is struck and then immediately muffled, and the same action simultaneously releases a sandbag that causes a long strip of paper to be drawn past the quill at constant speed. Instead of a needle, this is connected to an inked quill. Next to this is a conical trumpet, also pointing downwards into the water, with essentially an Edison phonograph pickup at the end. In the keel of the ship you mount a large bell poking through the hull, shaped to focus a sonar ping downwards. However, with hindsight, they could have done it, though it is pure speculation how well it'd work: Obviously people in the age of sail did not have the technology in its modern form, and prior to the 17th century they didn't even have the basic physics to understand the principle. Besides, they've been in use for centuries and centuries - I think the sailors might have gotten it right.
I think lead lines are honestly your best bet. not easy under even the best circumstances, let alone when you're on a rocking ship in the middle of the ocean. The ship itself is traveling through the water, and therefore affects the waves - your plane does not, presumably, unless it's essentially skimming the water.Waves are turbulent and travel in many different directions, leading to interference and complicated motion.Sure, you could attempt to measure the average wavelength of nearby waves, and try to characterize them (breaking, swells, etc.), and from there estimate the depth of the water. Waves do behave differently at different depths for instance, for a given wavelength, there's a particular depth beyond which a wave will not break. That should indeed be enough to map out most of the deeper regions of the continental shelf. Fortunately, longer lead lines have also been used, reaching as long as 50 fathoms (~91.4m), or 300 feet. This might not seem very helpful, as the continental shelf ends, on average, at depths of about 200 feet, though there's rather large variation - it can stretch up to around 300 feet in places. Shallow water lead lines were used to depths of about 20 fathoms (~36.6m, or about 120 feet, as 1 fathom = 6 feet). You're right, dropping a weighted rope is indeed a possibility - as you might already know from historical records - but it can actually operate in deeper water than you might imagine.
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