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<p class="noindent">We derive closed-form solutions for point-force generated motions in a continuously inhomogeneous half-plane, which represent the complete elastic wave-train in the interior domain obeying traction-free boundary conditions at the horizontal surface. More specifically, a special type of material inhomogeneity is studied, where the shear modulus varies quadratically with respect to the depth coordinate. Furthermore, the material density profile varies proportionally to the aforementioned profile, while Poisson’s ratio remains fixed at one-quarter. Limit forms for the Green’s functions are derived for both zero frequency and for the equivalent homogeneous medium. Next, a series of numerical results serve to validate this mechanical model, and to show the differences in the wave motion patterns developing in media that are inhomogeneous as compared to a reference homogeneous background. These singular solutions are useful within the context of boundary element formulations for the numerical solution of problems involving nonhomogeneous continua, which find applications in fields as diverse as composite materials, geophysical prospecting, petroleum exploration and earthquake engineering. </p>

Time-harmonic elastodynamic Green’s function for the half-plane modeled by a restricted inhomogeneity of quadratic type