EXACT SOLUTIONS FOR NONLINEAR WAVES ON THE SURFACE OF A DIELECTRIC LIQUID IN A TANGENTIAL ELECTRIC FIELD IN 3D GEOMETRY

As shown in [1], nonlinear plane waves may propagate without distortion along the surface of a dielectric liquid in a strong horizontal electric field. This situation occurs in ideal liquids with high permittivity. In this work, it will be shown that this unusual property can be generalized to the 3D case. Waves of arbitrary geometry (including those without planar symmetry) may propagate without distortion in or against the direction of the electric field, and the amplitude of a wave does not influence its velocity. This implies that the 3D equations of motion have exact solutions containing an arbitrary function of two variables that describes the initial wave profile [2]. The effect of nonlinearity on the propagation of waves along the surface of dielectric liquids in a tangential electric field (or along the surface of magnetic liquids in the magnetic field; these problems are mathematically equivalent [3,4]) was usually studied for the spectrally narrow wave packets in the approximation of a small surface inclination angle (see, e.g., [5–7] and references therein). This approach leads to a nonlinear Schrodinger equation for the complex wave amplitude. As will be shown below, the evolution of the surface of a liquid with a high permittivity in a strong external field may be effectively studied without the use of the envelopes. Let us consider the potential flow of an ideal dielectric liquid of infinite depth with a free surface in a homogeneous external electric field. In the unperturbed state, the liquid boundary is the 0z horizontal plane (the z axis of the Cartesian coordinate system is perpendicular to the liquid surface; the x and y axes lie in the surface plane). Let the electric field vector be directed along the x axis and the function ( , ,)x y t be the deviation of the boundary from the plane; i.e., the region occupied by the liquid is confined by the free surface z . The dispersion relation for the electrocapillary waves in the horizontal electric field has the form [3,4]: 2