My question is purely about the shape of the equipotential surface in 1 – or equivalently, and more graphically, about the difference in height between high tide and low tide if the Earth were entirely covered by an inertialess ocean with zero viscosity.
Wikipedia says "The theoretical amplitude of oceanic tides caused by the moon is about 54 cm" but gives no reference for this figure. ...
What I am looking for, therefore, is a source which can be considered authoritative and is not just a referenceless repeating of "what everyone knows".
1. The University of Hawaii has a website: "Exploring Our Fluid Earth" which they describe as:
"Exploring Our Fluid Earth is based on the nationally recognized Fluid Earth/Living Ocean (FELO) aquatic science curriculum (Klemm et al., 1990; Klemm et al., 1995). The Exploring Our Fluid Earth curriculum is grounded in the inquiry approach to learning and examines marine and freshwater systems of the earth by studying the influence of water on the planet.".
They explain in simple terms that the exact answer depends upon:
The location of the Moon, it's elliptical orbit and the lunar declination
The location of the Earth with respect to the Sun, it's elliptical orbit and the local geographic features:
"Factors that influence tidal range occur not only on a solar system scale, but also on local scales. Tidal ranges vary considerably at different points of a coastline due to seafloor features. When oceanic tidal bulges hit wide, shallow continental shelves the height of the tide is usually magnified. Conversely, mid-oceanic islands that rise steeply from the seafloor and do not have continental shelves have smaller tidal ranges. Mid-oceanic islands often have very small tidal ranges of 1 meter or less.
In narrow mouthed basins that are connected to the ocean, the tides often rise higher than in wide bays and harbors. This is analogous to pouring an entire can of soda into a short wide glass and a tall narrow glass. The soda will rise higher in the narrow glass, because there is not as much area to spread out as in the wide glass. The same amount of seawater will rise higher in a narrow basin than in a wide-mouthed harbor.".
2. The Public Encyclopedia Services Home Page has a webpage titled: "Ocean Tides - The Physics and Logic" which explains in detail (while certainly leaving some things out) the math behind the calculations.
3. The National Oceanic and Atmospheric Administration, U.S. Department of Commerce, has over a dozen webpages: starting with the Welcome page: "Tides and Water Levels", and while seemingly devoid of any math, the eleventh page, titled: "Tides Roadmap to Resources", lists numerous references.
4. PhysicalGeography.Net has a far too short explanation with great graphics, that depicts a greater range:
5. The Wikipedia webpage you referenced "Tides" is indeed a bit sparse on math too, try this Wikipedia page: "Tidal Acceleration".