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How is a transverse water waves set up in a pond and spread out in the form of ripples when a stone is thrown? In particular, how does the gravity, the surface tension and elasticity of water help in setting up the wave?

For a transverse wave to be supported by a medium, it should have a reasonably large shear modulus. Isn't it? Since fluids do not support shear, how does the water support transverse waves?

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The usual water waves are gravity waves that occur on the surface of a liquid due to the gravitational pull of deviations from the flat surface back to the mean surface level. The water particle motion is actually approximately circular and has longitudinal components. Transverse waves in solids need a shear modulus not gravity waves nor surface tension waves in liquids, which play a role at very small wavelengths and amplitudes.

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I think a reasonable question here, as well, is why the liquid (water) is so well confined in the first place? Gravity, of course, which is acting perpendicularly to the water is what holds it in that region, essentially creating a closed thermodynamic container.

I like to think throwing the pebble in as adding energy to the system. An increment of energy will cause the particles inside the system to gain some kinetic energy. And I think it is quite obvious that throwing a pebble into a lake does not dramatically increase the temperature of the water.

The fluid doesn't need to shear, as I think the most logical answer here is simply that the water molecules immediately in the vicinity of the pebble will impart energy directly to neighboring molecules. Obviously this is not enough energy to cause the molecules to change phase and evaporate, so they stay in the water phase and do not leave the surface entirely.

If you look at water droplets in slow motion you will likely find the answer. It's very much like jumping on a trampoline; pebble drops in, the body of water pushes upward with an equal force, and in doing so, carries many water molecules upwards, eventually breaking the surface of the water, sending water up through the point of entry, eventually crashing back down. This cycle repeats and the collisions end in a net result of the energy being distributed out into the x-y plane, as opposed to further down into the z-axis (think about the fact that if the energy went purely downward the temperature of the lake would have to increase by an amount of energy equal to the stone being thrown in- this is not the case as per what we observe).

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