Nash-Moser Theory for Standing Water Waves

We consider a perfect fluid in periodic motion between parallel vertical walls, above a horizontal bottom and beneath a free boundary at constant atmospheric pressure. Gravity acts vertically downwards. Suppose the underlying flow is two-dimensional in a vertical plane orthogonal to the walls and satisfies the constant-pressure condition on the free boundary where surface tension is neglected. Suppose also that it is symmetric about a plane midway between the walls, and periodic in time. Such motion, which can be extended to give a two-dimensional flow of infinite horizontal extent that is periodic in space as well as in time, is referred to as a standing wave. Unlike progressive (or steady) Stokes waves, standing waves are not stationary relative to a moving reference frame.