Do ocean waves have mass?

Consider a wave propagating across the surface of an ocean of water.

Clearly the medium (the water) in which the wave is travelling has
mass.

But would a physicist say that technically the wave itself has energy
but no mass? The reasoning being that mass is not being laterally
transported, but rather water molecules basically just oscillate in a
circle proportional to the wave size, or something like that.

Or in the case of water waves would the physicist say the wave has both
energy and mass?

Answers/opinions would be welcomed.

Thanks,
Rob

.

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