Modulational Instability of Nonlinear Capillary Waves on Thin Liquid Sheet

Modulational instability of weakly nonlinear capillary waves on a thin liquid sheet with no rigid boundary is theoretically investigated. By using the derivative expansion method, a simple nonlinear Schrödinger equation for the amplitude modulation is derived from the two dimensional Laplace equation for the velocity potential. It is found from this nonlinear equation that wave trains of finite amplitude are modulationally unstable. The phenomenon of the break-up of the sheet may be due to such an instability.