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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)?

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  • $\begingroup$ 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/… $\endgroup$ – CuriousOne Aug 2 '16 at 19:51
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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.

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