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While water moves up and down on the surface, does it move the same way below surface also? Or does it fade into a compression wave somewhere below the surface?

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Water wave is combination of traverse and longitudinal waves with a dispersion relation:

$$c=\sqrt{\left( \frac{g\lambda}{2\pi}+\frac{2\pi \gamma}{\rho \lambda} \right) \tanh \frac{2\pi h}{\lambda}}$$

and displacement profile $$\xi+\eta i=\frac{ga}{kc^{2}} \frac{\cosh[k(y+h)+i(kx-\omega t)]}{\cosh kh}$$

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In deep water, the particles move in more circular way, and the amplitude decays with depth. No perceptible motion deeper than $\dfrac{\lambda}{2}$.

In shallow water, the particles move in more elliptical manner, and they're pure longitudinal at the bottom.

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See also another answer here

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Individual points in the water move in circles, up, forward, down, back. Collectively, this forms wave motion.

The magnitude of the motion is greatest at points on the surface, and the magnitude of the motion decreases with depth. Beyond about a half wavelength below the surface, there is no motion.

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