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According to wikipedia, the restoring force of water waves are gravity or surface tension, depending on the wave length and water depth. I can't understand exactly the gravity explanation.

If gravity was the sole restoring force, the result could be a local temporally oscillatory movement, like a pendulum.

I can't see it as a source of a propagating wave.

There are compressive waves as sound, or seismic waves. Or tensile waves that happens in a string or a membrane.

In all that cases, it can be understood why an initial movement propagates.

I understand water waves as a side effect of pressure waves, but I'm not sure if it is right because they are not usually explained that way.

For example: a stone is thrown in a lake. There is a local sudden pressure increase. As a consequence, the level of water increases near the event, (because the atmospheric pressure is constant). The pressure wave in the water propagates to all directions, including upwards, leading to concentric circles that we see spreading at the water surface.

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    $\begingroup$ Pressure waves travel in water at the speed of sound (about 1531 m/s in sea water). Water waves travel slowly at a speed of $c=\sqrt {g/k}$ for deep water and $c=\sqrt {gh}$ for shallow. The basic model of water waves does not contain pressure at all - see en.wikipedia.org/wiki/Boussinesq_approximation_(water_waves) $\endgroup$ – Alex Trounev Jan 11 at 11:20
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The gravity scenario is not that hard to grasp, if you have the right example. I'll try to provide one here.

We begin with a large container of water and drop a bowling ball into it from a large height. The ball strikes the water surface and as it descends into the water, it pushes the water in front of it sideways to get it out of the way.

Now we have water pushed into motion radially outward from the impact point of the bowling ball. The non-moving water immediately in front of this moving water has inertia, and resists being accelerated by the moving water. But the moving water continues pushing and the result is that it "piles up" against the nonmoving water and forms a high spot in the water surface.

The nonmoving water is reluctantly pushed into motion radially outward and at the same time, gravity begins pulling back down on the high spot. The water entrained in the high spot falls downward and then sideways, with some water moving radially outwards and some inwards. We then see the high spot in the water propagate radially outward and another "lump" radially inward.

Those moving high spots are waves, and the water in them is bouncing not only up and down but sideways as well. Looking at the wave in cross-section, an individual parcel of water traces out a circular path as the wave passes by.

There are animations of gravity waves on the web that show this process quite clearly; I'll link one here in an edit if I can remember where to find my favorite.

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