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The Cellular Automata (CA) method is based on the idea that the macroscopic behavior of a system can be captured by using simple local rules running at a microscopic level. In other words, a system can be modeled by means of simple local rules that govern the behavior of the whole system. In this thesis a local CA rule set is introduced and a methodology is developed to model physical systems that are governed by one and two dimensional wave equations. One dimensional systems are also successfully modeled by using CA and FEM techniques working as coupled, whereas two dimensional systems could only be modeled in an error margin due to the variation of the introduced time scaling factor when external forces are involved. Also, the applicability of the CA method to fracture mechanics problems is investigated.