A theoretical study presents an analytical solution for the transient response of a cylindrical cavity embedded in a three-dimensional elastic medium subjected to impact loading. The result provides an exact closure to a problem that has so far been tackled mainly with numerical methods, and offers a benchmark against which finite element simulations can be checked.

The question of how an elastic wave propagates around a buried cavity is a classic one in solid mechanics and in seismic and geotechnical engineering. The cylindrical geometry in three dimensions is more difficult than the two-dimensional case, because the displacement field depends on all three coordinates and on the reflection of waves at the cavity surface. The analytical formulation makes it possible to separate the response into identifiable contributions and to follow their time evolution without discretising the domain.

The method builds on linear elasticity theory and on integral transform techniques to solve the equations of motion with the appropriate boundary conditions at the cavity wall. The impact load is imposed as a time-dependent initial or boundary condition, and the solution is reconstructed in the time domain from the transformed response. The derivation yields closed-form expressions for the quantities of interest, such as displacements and stresses as a function of time and position.

Having an exact solution is useful for validating numerical codes used to predict vibrations induced by blasting, drilling or earthquakes near tunnels and underground cavities. It also allows a systematic exploration of how the response changes when varying parameters such as the elastic properties of the medium, the cavity radius or the time profile of the impact, without repeating costly simulations. The work follows the line of obtaining reference solutions for wave-structure interaction problems in continuous media.