Can a water wave be a longitudinal wave?
Of course. Sonars, two people "talking" to each other underwater, knocking on top of a can of soup, knocking on ice, when fish bump into each other etc.
Transverse water waves are mostly apparent on the surface and during tsunami-like events asaik. Moving a water molecule up and down should cause a transverse wave along the normal plane, a longitudinal one along its oscillating axis, and a combination of both at 0 < angle < 90 degrees.
Longitudinal wave forms in water other than the sound wave option, might possibly be induced by spatially varying surface tension.
Surface waves are a combination of longitudinal and transverse motion. There is significant longitudinal movement in shallow water waves (where the ratio of wavelength/depth is significant). Tsunamis cause a huge longitudinal movement of water, as we all know.
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