The Secondary Effect of the Seismic Waves on Large Urban Agglomerations

A set of pointlike inhomogeneities placed on the flat surface of an elastic half-space is put forward as a model for large urban agglomerations, consisting of buildings, constructions, erected on a limited area on Earth’s surface (e.g., cities). Under the action of an incident plane seismic wave a hierarchy of scattered waves arising from these inhomegeneities is identified, and their effect on the pointlike inhomogeneities is analyzed. We show that in the long-wavelength limit the first-order scattered waves are of Coulomb type, i.e. the displacement in these waves goes like the inverse distance from the scattering centres. We estimate the displacement produced by the scattered waves in this case, and find that it is large for large and agglomerated buildings, constructions, placed over large distances, i.e. precisely for what we call "large urban agglomerations". In the short-wavelength limit the Coulomb-like dependence acquires an oscillating factor. In this case we identify a divergent displacement produced by the scattered waves when the incident wave propagates parallel to the surface; leaving aside this special case, a necessary condition for the convergence of the series produced by the hierarchy of the scattered waves is a rare (sparse) distribution of small-size inhomogeneites.