"In this dissertation we study the so-called “city effect” problem. This effect occurs when earthquakes strike large cities. In earlier studies, seismic wave propagation was evaluated in a separate step and then impacts on man made structures above ground were calculated. The 1985 Michoacan earthquake in Mexico City led Wirgin and Bard (1996) to hypothesize that city buildings may collectively affect the ground motion during an earthquake. Ghergu and Ionescu (2009) proposed a model of this phenomenon and a solution algorithm. Our contribution is to extend their work and to provide a mathematical analysis for proving the existence of preferred frequencies coupling vibrations of buildings to underground seismic waves. Given the geometry and the specific physical constants of an idealized two dimensional city, Ghergu and Ionescu computed a frequency that will couple vibrating buildings to underground seismic waves. This frequency was obtained by increasing the number of buildings at the expense of solving larger and larger systems. Our idea is to use a periodic Green's function and perform computations on a single period. That allows for much faster computations, and makes it possible to consider more complex geometries within a single period. We provide a rather in depth and proof based account of different formulations for the periodic Green's function that we need. We show that they are indeed fundamental solutions to the Helmholtz operator and we analyze their convergence rate. Finally, we give a mathematical proof of existence of preferred frequencies coupling vibrations of buildings to underground seismic waves."