An approach to studying nonstationary wave processes in an elastic half-plane with mixed boundary conditions of the fourth boundary-value problem of elasticity is proposed. The Laplace and Fourier transforms are used. The sequential inversion of these transforms or the inversion of the joint transform by the Cagniard method allows obtaining the required solution (stresses, displacements) in a closed analytic form. With this approach, the problem can be solved for various types of loads Keywords: elastic half-plane, stress state, nonstationary waves, integral transforms, mixed boundary conditionsIntroduction. There are four types of boundary-value problems in elasticity (see, for example, [9]): (i) the stress vector is given (first boundary-value problem); (ii) the displacement vector is given (second boundary-value problem); (iii) the normal component of the displacement vector and the tangential components of the stress vector are given (third boundary-value problem); (iv) the normal component of the stress vector and the tangential components of the displacement vector are given (fourth boundary-value problem). The first two types are main conditions, and the last two conditions are mixed. Whether the analytic solution of nonstationary problems can be found depends on the type of boundary conditions. Nonstationary wave processes in an isotropic elastic half-space (half-plane) under concentrated or distributed surface loads were subject of many studies (see, for example, [3,6,7,11] and the references therein). To solve corresponding boundary-value problems, use is made of effective analytic and combined numerical/analytical methods that employ the Laplace transform with respect to the time coordinate and the Fourier (Hankel) transform with respect to the linear coordinate. The chief difficulty is the recovery of the original functions, which depends, primarily, on the space-time distribution of load over the boundary and the type of boundary conditions. Studies usually focus on separate aspects of wave processes: asymptotic construction of the displacement and stress fields near wavefronts [26,27]; the displacements of particles of the surface of the half-space and the symmetry axis [15,25] or at a considerable distance from it [15]. The joint transform is inverted using the Cagniard technique [8,12] and taking into account the homogeneity of the joint transform in the transform parameters. Solutions for the displacements and stresses inside a half-space under a uniformly distributed step load were found in [14,[21][22][23] using the residue theory to invert the Laplace transform. The expressions derived in [10] are the superposition of the analytic solutions of the Lamb problem and the convolution with respect to the radial coordinate of the load distribution function and the fundamental solution. Two types of spatial distribution of load were considered and the elastic displacements resulting from a load varying with time as a delta function were calculated. This resulted in discontinuity of the calculated dis...
