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We rigorously analyze the SH (shear horizontal) elastic wave diffraction for the finite crack located between the plane interface of the slab and the half-space medium. The incident field is taken as a plane wave. Introducing the Fourier transform of the scattered field and applying boundary conditions in the transform domain, the problem is formulated in terms of the Wiener-Hopf equation. The Wiener-Hopf equation is solved exactly via a factorization procedure leading to the formal solution. An approximate solution of such an equation is proposed.