Bernoulli's theory of the equilibrium tide could explain the tidal periods and the variation between spring tide and neap tide (see Lecture 20). It could not explain why the tidal range is so much larger at some places than at others.
Laplace noticed that the basic premise of the equilibrium theory - that high and low water always follow the position of the moon instantly - was unrealistic, as it would involve the movement of large amounts of water around the Earth at the speed of the moon's movement. He considered the tides the response of a number of closed basins (oceans, nearly enclosed seas, and bays) to a global forcing mechanism, similar to the reaction of fish tanks standing on a table when the entire table is moved:
If the table is moved very slowly, the water in the fish tanks will barely move. If the table is moved extremely fast, the water in the fish tanks will show some small vibration but will not be able to follow the movement much. But if the table is moved at the right frequency, resonance will occur, and the water in the tanks will slosh backwards and forwards and possibly spill over.
Just as organ pipes of different size produce different audible frequencies, fish tanks of different size resonate at different frequencies. In the diagram above the tank on the right has a longer resonance period than the tank on the left. The response of ocean basins of different size to the tidal forcing will therefore vary and depend on how close the tidal forcing frequency comes to the resonance frequency of each ocean basin. This explains why some regions have a larger tidal range than others.
Bays and small semi-enclosed seas can experience resonance with the tide of the adjacent ocean, which produces much higher tides in the bay than in the deep ocean:
If the tidal amplitude in the deep ocean is known, the Laplace transform allows the calculation of the tide in the adjacent bay. (The red balls indicate the water movement at the mouth and at the head of the bay.
Illustration and animation: © M. Tomczak, Creative Commons license