Acoustic wave vector and acoustic impedance

I'm working on Perfectly Matched Layer simulation, and I got this question.

As far as I know, generally, acoustic phase velocity is $$v=\frac{\omega}{k}=\sqrt{\frac{C}{\rho}}$$ where $\omega$ is angular frequency, $k=\frac{2\pi}{\lambda}$ is wavevector, C is Elastic constant, $\rho$ is density of medium.

However, in the "Parameter Extraction and Support-Loss in MEMS Resonators",

http://arxiv.org/abs/1304.7953

the acoustic wavevector is [eq. 19]

$$k=\frac{\omega\rho}{Z}$$

where Z is acoustic impedance of medium, and

$$Z_{compress}=\sqrt{\frac{\rho E (1-\nu)}{(1+\nu)(1-2\nu)}}$$ $$Z_{shear}=\sqrt{\frac{\rho E }{2(1+\nu)}}$$

Why is this different??

Is there any textbook which explains below equation (eq. 19)?

• The details for wave motion in elastic solids are usually spelled out in books on continuum mechanics. For a very brief intro to elasticity in this case see en.wikipedia.org/wiki/… Aug 2, 2016 at 19:51

There is no difference. As the acoustic impedance of the medium is defined as $Z=\rho v$, the first part of eq 19 is just $k=\frac{\omega}{v}$ with $v=\frac{Z}{\rho}$
The formulas for acoustic impedance result from the definition, replacing the speed with formulas expressing the speed of sound in a solid medium (longitudinal and transverse waves). The formula $v= \sqrt{\frac{C}{\rho}}$ is valid for longitudinal waves propagating in a rod, when C is the Young's modulus.