Analytical solution of the asymmetric transient wave in a transversely isotropic half-space due to both buried and surface impulses

• A transversely isotropic half-space under arbitrary time dependent tractions is considered.
• A potential method and joint Laplace–Hankel integral transforms are used to find analytical solution for the problem.
• The solutions presented in the integral forms, are numerically evaluated for a synthetic material.
• The results are used as kernels in integral based solution.

With the aid of a complete set of two scalar potential functions, the problem of transient wave propagation in transversely isotropic half-space, subjected to time dependent tractions applied on a finite patch at an arbitrary depth below the free surface of the half-space is investigated. With the use of the displacement–potential function relationships in a cylindrical coordinate system, the coupled equations of motion are uncoupled; resulting in two separate partial differential equations one of which is second order and the other is fourth order. These two partial differential equations are solved with the aid of both Fourier series expansion and joint Hankel–Laplace integral transforms. The solutions are also investigated in details for tractions varying with time as Heaviside step function, which may be used as a kernel in any integral based method for more complicated elastodynamic initial-boundary value problems. Moreover, some displacement Green׳s functions are numerically evaluated for a synthetic transversely isotropic material to graphically demonstrate the transient motion of the free surface of the half-space.