Water waves induced by a fluctuating tangential stress

In the framework of linear wave theory both the effects of tangential and normal stresses on the water waves are discussed without the assumption of irrotational water motion. A formal solution initially at rest with level surface and developed under the actions of the surface stresses depending arbitrarily on time and sinusoidally on space is obtained.A progressive wave type tangential stress whose wave number and frequency are satisfying the dispersion relation of the water waves is shown to be equivalent to the normal stress of the same type on the growth of the waves except the phase relations between the stresses and the water motion. The growth rate of the waves induced by the tangential stress is also shown to be quite insensitive to the actual value of the viscosity.The rotational part of the water motion can dominate only in the early stage of wave generation and becomes negligible with the growth of the waves relative to the irrotational part of the motion even in the case where the motion is induced by the tangential stress alone. Therefore it is not reasonable to neglect effect of the tangential stress on wind waves even if the developed wind waves seem to be irrotational.