The diffraction of flexural waves by a short straight crack in an elastic thin plate is considered. The vibrations of the plate are described by the Kirchhoff model. The Fourier method transforms the problem to integral equations of convolution on an interval. The theorems of existence and uniqueness of solutions for these equations are proved. The numerical procedure is based on the orthogonal polynomials decomposition method. It leads to infinite systems of algebraic equations for the coefficients. The truncation method is proved to be applicable to these systems due to the special choice of the polynomials. A physical interpretation of numerical and asymptotic results obtained for the directivity of the scattered wave and for the stress intensity coefficients near the ends of the crack is suggested.