On the characteristic time for the realization of instability on a flat charged liquid surface

A nonlinear integral equation that describes the time evolution of the amplitude of a nonlinear unstable wave on the flat uniform charged surface of an ideal incompressible liquid has been derived and solved. The characteristic time for the realization of instability is found to be determined by the initial amplitude of a virtual wave initiating the instability and the supercritical increment in the Tonks-Frenkel parameter. At a zero supercritical increment, the characteristic time for the realization of instability is only determined by the initial amplitude and can be rather long (up to eight hours). This effect is characteristic of a flat charged liquid surface and does not occur in charged drops.