Problems about vibrations of an orthotropic layer with a cylindrical cavity with an arbitrary transverse cross section under the action of a load applied on its surface are considered. In the case of cavities with a small relative size, an asymptotic approach to the calculation of fields is proposed. An estimate of the applicability area of the asymptotic approach in comparison with the method of boundary integral equations, as well as a comparison with the solution obtained based on the Born approximation are presented.