The dynamics of thin sheets of fluid II. Waves on fluid sheets

Abstract

It is shown that capillary waves are of two kinds, symmetrical waves in which the displacements of opposite surfaces are in opposite directions, and antisymmetrical waves in which the displacements are in the same direction. Any disturbance can be regarded as composed of these two types of wave. The antisymmetrical waves are non-dispersive. In a sheet of uniform thickness a moving point disturbance produces two narrow line-like waves. In a radially expanding sheet a fixed disturbance point produces two narrow disturbances in the form of cardioids. It is shown theoretically that a finite change in direction of flow can occur at a cardioid which therefore assumes the form of a sharp edge. A method was found for producing and photographing a sheet with a sharp edge in the form of a cardioid. The symmetric waves are very different, they are highly dispersive and are propagated much more slowly than the antisymmetrical waves. Experimentally a point disturbance produces both kinds of wave simultaneously. Reflexion photographs show the antisymmetrical waves, while the schlieren method is needed to reveal the symmetrical waves. The symmetrical waves produced in a moving sheet by a point disturbance are parabolas when the sheet is uniform in thickness, and of a more complicated form when the sheet is expanding. The predicted wave patterns agree with those revealed by the schlieren photographs.