In this chapter we treat plane waves specified by a wave normal and a particle motion vector . Two types of waves, longitudinal waves and shear waves, are observed in solids. For low symmetry directions, there are generally three different waves with the same wave normal, a longitudinal wave and two shear waves. The particle motions in the three waves are perpendicular to one another. Only longitudinal waves are present in liquids because of their inability to support shear stresses. The transverse waves are strongly absorbed. Acoustic wave velocities (v) are controlled by elastic constants (c) and density (ρ). For a stiff ceramic (c ∼ 5 × 1011 N/m2) and density (ρ ∼ 5 g/cm3 = 5000 kg/m3), the wave velocity is about 104 m/s. For low frequency vibrations near 1 kHz the wavelength λ is about 10 m. The shortest wavelengths are around 1 nm and correspond to infrared vibrations of 1013 Hz. Acoustic wave velocities for polycrystalline alkali metals are plotted in Fig. 23.2. Longitudinal waves travel at about twice the speed of transverse shear waves since c11 > c44. Sound is transmitted faster in light metals like Li which have shorter, stronger bonds and lower density than heavy alkali atoms like Cs. The tensor relation between velocity and elastic constants is derived using Newton’s Laws and the differential volume element shown in Fig. 23.3(a). The volume is equal to (δZ1) (δZ2) (δZ3). Acoustic waves are characterized by regions of compression and rarefaction because of the periodic particle displacements associated with the wave. These displacements are caused by the inhomogeneous stresses emanating from the source of the sound. In tensor form the components of the stress gradient are ∂Xij/∂Zk and will include both tensile stress gradients and shear stress gradients, as pictured in Fig. 23.3(b). The force F acting on the volume element is calculated by multiplying the stress components by the area of the faces on which the force acts.
Origin of Piezoelectricity in Piezoelectric Ceramics from the Viewpoints of Elastic Constants Measured by Acoustic Wave Velocities
