We present several sensitivity tests, that were applied to exhibit the effects of earthquake source rupture characteristics on amplitudes, frequency contents and arrival times of earthquake generated tsunami waves in the far field, as case studies in the eastern Mediterranean. The investigated earthquake parameters are principally epicentral location, focal mechanism parameters (strike, dip and rake angles), faulting area dimensions, maximum displacement and focal (centroid) depth. We have implemented a numerical method of TUNAMI-N2 based on non-linear shallow-water theory to obtain synthetic water surface fluctuations at selected pseudo tide gauge locations in the eastern Mediterranean. It has been observed that the most important source parameters that effect tsunami wave characteristics in the far field are:  magnitude and seismic moment (M-o = mu x A x D) of earthquake that is a measure of the energy release radiated at the centroid depth. We have observed that wave amplitudes and shapes change considerably with variation of magnitude and seismic moment since tsunami waves develop in direct proportional relation to them;  another parameter is the accurate estimation of tsunamigenic earthquake epicentre. Variation of the earthquake location does not significantly affect the initial tsunami wave heights, but final tsunami wave characteristics and their arrival times have been slightly changed due to the variation of distance between the epicentre and coastal plains along the path. Especially, wave spreading causes tsunami waves to decrease in amplitude as they move away from earthquake source;  variation in focal mechanism solutions modify the tsunami wave propagation directions, wave amplitudes, shapes and arrival times of tsunami waves observed at the coastal plains;  in addition, due to the linearity between the amount of vertical co-seismic displacement and initial tsunami wave, very different tsunami amplitudes were obtained at each pseudo tide gauge stations in case of the variation in maximum displacement;  details of local bathymetry (e.g., extended sedimentary shelf area) and the sea bottom irregularities (e. g., sea-mounts, volcanoes, accretionary prisms, trenches, pressure ridges) clearly have crucial effects on tsunami wave characteristics in the far field. Historical records confirm that the eastern Mediterranean region is at risk from tsunamigenic sources located on the Hellenic-Cyprus arcs. Thus, higher resolution near-shore bathymetry data as well as a detailed study of potential tsunami sources in segments of subduction zones are necessary to verify our simulation results.