Tide in an enormous hypothetical canal

[granpa]

imagine a man made canal many km wide and many km deep running the full length of one hemisphere at the equator. let one end of the canal be represented at the origin and the other at x=2π.

the strength of gravity at any point due to the Earth plus tidal forces of the moon should equal a constant plus (I assume) a full sine wave. given enough time the surface of the water should form a sine wave. but as the moon moves across the sky the sine wave representing the force of gravity should move in one direction (at a speed less than the speed of sound in water). if the canal formed a full circle all the way around the Earth then the water would simply follow this sine wave. but because it only stretches across one hemisphere its obvious that the water can't possibly follow the sine wave. so what does the water do. how would I solve this?edit:actually maybe it would just follow the sine wave. the water at each point just moves back and forth so I guess it could.

 

[Naty1]

There are standard ways that have been developed to calculate tidal movement...I tried a GOOGLE on "tidal Forces" and one explanation of many looking conceptually interesting was...

http://csep10.phys.utk.edu/astr161/lect/time/tides.html

Diurnal tides (two highs, two lows daily) here along the east coast of the US occur approximately an hour later each day due to the rotation of the moon. Tides in other parts of the world...and maybe the equator...may follow different patterns. Also the patterns vary by time of year in a given location; Around Jan or Feb in the NY CT area tides are "higher highs" and if they coincide with storms, severe flooding is possible.

I suspect your problem has many answers depending on the accuracy desired...


Also, You may want to check NOAA websites for tidal formulations...

 

[granpa]

after further reading it appears that it would indeed form standing waves after all. if there were no friction each part would try to rise as the moon passed overhead and sink as it moved away. if there were friction it would move out of phase slightly.

 

[granpa]

so I guess, roughly speaking, you could think of the nodes as capacitors and the antinodes as inductors. and tidal forces as an ac voltage.

 

[Naty1]

Your post #3 appears practical; I thought you were seeking actual mathematical formulas to solve your puzzle...I have seen tidal formulas over the years, as I noted, I believe NOAA has some posted...