1) The water wave is a surface wave, does it mean only the molecules on the surface is undergoing wave motion?

2) I know water is hard to compress, , some water molecules will displace to sideways in a trough. However, I don't understand the following statement in my lecture notes:

"Picture the point on a wave between the trough and the crest (i.e. where the y displacement is zero). The slope here is greatest, which suggests that the longitudinal displacement will be at a maximum (most water rearranged)."

I don't understand the relation between slope and longitudinal displacement and I feel hard to understand why there is a maximum longitudinal displacement at this point, can you explain more?

Thank you for taking time to look at my question.

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1. No, water molecules below the surface do move. Water molecules in a "surface wave" move in circular orbits. The diameter of the orbit decreases with distance below the water surface. At a verical distance of one wavelength below the surface, the orbit becomes nearly zero, i.e. water molecules stop to move at such depth and below.

2. As said above, water molecules move in circular orbits in a water wave. A circular orbit is indeed a transverse wave and a longitudinal wave superimposed on each other. Hence, water molecules have both vertical and horizontal displacements. Note that these two displacements are 90 degrees (or a quarter of wavelength) out of phase.

Because the logitudinal (i.e. horizontal) motion of molecules and the transverse (i.e. vertical) motion of molecules are 90 degrees out of phase. At the wave crest and trough, the vertical displacement is maximum, the horinztonal displace is zero. Hence, at the mid-point between the crest and trough (i.e. the y-displacement, which indicates the vertical displacement, of the wave is zero), the horizontal displacement, which has a phase difference of 90 degrees with the vertical displacement, will be at maximum.

Perhaps we can look it on another way using Simple Harmonic Motion (SHM) of water molecules. Just consider the longitudinal motion of the water molecules. These molecules perform Simple Harmonic Motion, the the equilbrium positions are located at the trough and crest of the water wave. Similarly, for the transverse motion of molecules, but the equilibrium positions are at the mid-point between the crest and trough.

Because the slope on the shape of the water wave indicates the speed of the water molecules on the transeverse (vertical) motion of molcules, the slope is greatest at the mid-point between the crest and trough. Thus, the vertical speed of molecules is maximum at this point. The horizotnal speed of molecules is thus minimum (i.e. zero) at such point, because the two motions are 90 degrees out of phase. From theroy in Simple Harmonic Motion, a zero velocity implies that the displacement is masximum.

The following web-page gives a very clear explanation on water waves. You may take a look on it:

http://labman.phys.utk.edu/phys221core/modules/m12...