The displacement field scattered by a rectilinear thin crack of finite length in a plate vibrating in bending is investigated. The boundary-value problem is reduced to integral equations on a segment by methods analogous to those developed in /1/. These integral equations are later replaced by the method of orthogonal polynomials, by infinite algebraic systems solved by the method of reduction. These systems also enable one to find the asymptotic form of the scattered field in the case of a short crack. The asymptotic form of the radiation pattern of a cylindrical wave diverging from the crack and the effective scattering cross-section are constructed. The results are monitored by using an optical theorem /2/.