The scattering of waves by an infinite elastic plate, which covers an acoustic half-space and is weakened by a crack in the form of an infinitely long cut of finite width with parallel edges is considered. The scattered field is expressed in terms of the solution of an integro-algebraic system of equations on the crack. The logarithmic characteristic of the kernel enables Bubnov's method with a basis containing Chebvshev polynomials of the first kind to be used for the numerical analysis. Particular attention is given to the asymptotic investigation of the scattering diagram and the amplitudes of the surface waves for a narrow crack and a thin plate. A comparison with the well-known model of a point crack enables the range of parameters of the problem where the point model is applicable to be indicated.