Propagation of nonstationary boundary disturbances in a half-plane filled with a homogeneous isotropic elastic medium

This study presents a method for addressing dynamic issues using an elastic half-plane subjected to different sorts of boundary disturbances. The motion of the half-plane is governed by wave equations that pertain to the scalar and non-zero components of the vector elastic displacement potentials. It is assumed that the starting circumstances are zero. An explicit solution to the problem is obtained by using the integral relation between the components of displacements and stresses of the half-plane boundary. This relation is expressed as a two-dimensional convolution with the influence function resulting from the principle of superposition. The properties of the convolution operation in two variables and the theory of generalized functions are used to derive the solution in integral form. Simultaneously, the acquisition of this solution relies on the technique of decomposing the influence function, whereby it is expressed as the multiplication of two components that meet the predetermined essential criteria. Hence, to get conclusive outcomes, it is important to factorize the impact function that has the given characteristics. The process of obtaining the necessary factorization of the influence function relies on describing its image as a multiplication of individual elements. The first derivation of this function was accomplished by the use of the joint inversion technique of the Fourier-Laplace transform, relying on analytical representations of pictures. Consequently, explicit integral formulae were derived to solve the issue and enable the determination of unknown displacements and stresses at any speed range of motion of the boundary conditions interface point. An example of a particular sort of boundary condition is provided to demonstrate the technique for addressing common situations.