**7. References**

Axmann, W. & Kuchment, P. (1999). An efficient finite element method for computing spectra of photonic and acoustic band-gap materials. J. Comput. Phys. Vol. 150, pp. 468, ISSN 0021-9991

Fig. 18. The band gap widths of the rectangular lattices with different rotating angles of

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

The authors thank the National Science Council (NSC 97-2218-E-150-006, 98-2221-E-150-026,

Axmann, W. & Kuchment, P. (1999). An efficient finite element method for computing

spectra of photonic and acoustic band-gap materials. J. Comput. Phys. Vol. 150, pp.

band gaps of elastic/acoustic modes in special phononic band structures.

and 99-2628-E-150-001) of Taiwan for financial support.

reticular geometric structures

**5. Conclusion** 

**6. Acknowledgment** 

468, ISSN 0021-9991

**7. References** 


**2** 

*1Russia 2,3Poland* 

**Topological Singularities in Acoustic Fields** 

 *1A.V. Shubnikov Institute of Crystallography, Russian Academy of Sciences, Moscow,* 

The influence of energy dissipation on the properties of bulk elastic waves in crystals is not at all reduced to trivial decrease in their amplitudes along propagation. In anisotropic media the situation is much more complicated than it looks like at first glance, at least for such specific directions of propagation as acoustic axes. The latter are defined as directions **m**<sup>0</sup> along which a degeneracy of the phase speeds of two isonormal waves occurs (Fedorov, 1968; Khatkevich, 1962a, 1964). The corresponding points of the contact of the degenerate sheets of the phase velocity surface *P* may be tangent or conical (Alshits & Lothe, 1979;

> 1 *v* 2 *v* 3 *v*

t **m**0

Fig. 1. Schematic plot of the section fragment of the three sheets of the phase velocity surface

Taking into account that formally the wave attenuation may be described as an imaginary perturbation of the phase speed, one could expect due to the damping either a shift or a split of the acoustic axis, of course if it is not created by a symmetry. As we shall see below, for an acoustic axis of general position it is just splitting what is realized, and with quite a radical transformation of the local geometry of the phase velocity surface. The other possible reason for sensitivity of the wave properties to a small attenuation is related to a polarization aspect. Indeed, it is known (Alshits & Lothe, 1979; Alshits, Sarychev & Shuvalov, 1985) that the acoustic axes indicate on the unit sphere of propagation directions <sup>2</sup> **m** = 1 the singular points in the vector fields of polarizations which are characterized by the definite vector

= 1, 2, 3) containing one tangent and two conical points of degeneracy

c **m**0

**1. Introduction** 

( ) *<sup>α</sup> v* **m** (α

Alshits, Sarychev & Shuvalov, 1985) (Fig.1).

**due to Absorption of a Crystal** 

*3Kielce University of Technology, Kielce* 

V. I. Alshits1,2, V. N. Lyubimov1 and A. Radowicz3

*2Polish-Japanese Institute of Information Technology, Warsaw,* 

