**5. Conclusion**

This chapter examines and discusses the acoustic waves in homogeneous medium and inhomogeneous medium, periodic structures with two media and one medium with geometrical periodicity. The wave velocities of shear and longitudinal modes in an isotropic material and those of quasi-SV, quasi-SH, and quasi-L modes in an anisotropic material are obtained using the finite element method. This method also discusses the tunable frequency band gaps of bulk acoustic waves in two-dimensional phononic crystals with reticular geometric structures using the 2D and 3D finite element methods. This study adopts the finite element method to calculate dispersion relations, avoiding the numerical errors, Gibbs phenomenon, from the PWE method. Results show that changing the filling fraction, scale a, and the rotating angles of unit lattices in the reticular geometric structures can increase or decrease the elastic/acoustic band gaps. The effect discussed in this chapter can be utilized to enlarge the phononic band gap frequency and may enable the study of the frequency band gaps of elastic/acoustic modes in special phononic band structures.
