Subject of this dissertation is presentation of a new phononic crystal structure to guide elastic waves in arbitrary angles and investigation of effects of structural defects. A new algorithm for studying of elastic wave propagation in the phononic crystals has been presented at first. In contrast with conventional finite difference time domain method, by using constitutive equations and strain-displacement relations, elastic wave equations have been derived based on displacement. Then, these displacement based equations have been discretized using finite difference method. By this technique, components of stress tensor can be removed from the updating equations. The new algorithm is called displacement based finite difference time domain method(DBFDTD). Since the proposed method needs less elementary arithmetical operations, its computational cost is less than that of the conventional FDTD method. Several numerical examples were computed with this method and the results were compared with experimental measurements and conventional FDTD method. The computational cost of the new approach was compared with conventional FDTD method as well. The comparison showed that the calculation time of the DBFDTD method is 37.5 percents less than that of the FDTD method. Afterward, a new heterostructure phononic crystal has been introduced. The new heterostructure is composed of square and rhombus phononic crystals. Presented heterostructure analyzing showed that the band gap can be extended by using the new structure. Effect of rhombus angular constant on the band gap of the heterostructure was studied too. This investigation showed that the band gap of the heterostructure phononic crystal would be wider than those of other heterostructures studied if the angular constant of the rhombus lattice is 80 degree. In addition, the ability of the presented structure in the arbitrary angle guiding of elastic waves was investigated. This study showed that by creating line defects in the composition of the square and triangular lattices, 30 degree bend waveguide can be realized. This bend guides elastic waves in the frequency range of 94-129 kHz. Also the study showed that by creating line defects in the composition of the square and rhombus lattices, 40 (20) degree bend waveguide could be obtained if the angular constant of the rhombus lattice is 80 (40) degree. The 40 (20) degree bend waveguide leads elastic waves in the frequency range of 121-127 (100-106) kHz. Also analyzing of the presented heterostructure showed that the structure can guide elastic waves in the arbitrary angle greater than 90 degree. Afterwards the effects of the cavity on the transmission sepectra of the presented waveguide was studied. Analysing of coupling effects in joined parallel square and triangular phononic crystal waveguides was another subject of this dissertation. The results showed that the beat phenomenon of the two-waveguide system is poor as well as the coupling effects. Finally an experimental sample of the presented strucuture (periodic arrangement of steel rods in the polyester host) was manufactured and tested. Comparison of the experimental transmission spectra with calculated one indicates that the experimental and numerical results are in good agreement. Key words: phononic crystal heterostructure, band gap extension, finite difference time domain, displacement-based formulation, arbitrary angle waveguide, structural defects, joined waveguides