The wavemaker problem

Water waves are generated by the oscillatory motion of a piston (note that the vertical scale of the free-surface motion is much exaggerated compared with the piston motion and the depth of the fluid).

The series solution for this problem contains a plane wave travelling to the right and an infinity of evanescent modes that decay rapidly with distance from the piston. The evanescent modes make a significant contribution to the fluid motion only in the region very close to the piston.