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The motivation for our studies is coming from simulation of earthquakes, that are modelled by elastic wave equations. In our paper we focus on stiff phanomenons for the wave equations. In the course of this article we discuss iterative operator splitting methods for wave equations motivated by realistic problems dealing with seismic sources and waves. The operator splitting methods are well-known to solve this kind of multidimensional and multiphysical problems. We present the consistency analysis for iterative methods as theoretical background with respect to the underlying boundary conditions. From an algorithmic point of view we discuss the the decoupling and non-decoupling method with respect to the eigenvalues. We verify our methods with test examples for which analytical solutions can be derived. Multidimensional examples are presented for realistic applications for the wave equation. Finally we discuss the results. ; Peer Reviewed