The Use of Vibration Principles to Characterize the Mechanical Properties of Biomaterials - Biomaterials: Physics and Chemistry - page 302

dilatational motion of the liquid (Herzfeld and Litovitz, 1959). Thus, a simplified form can
be used to obtain the complex wave number ( k ) and ultimately the attenuation () from
which the fluid viscosity can be extracted. The complex wave number is given by the
following equation:

ˆ kk 

  (5)

1



where the real number k 1 is calculated as k  and  is the attenuation of the wave due

1

C

1

to the viscosity of the testing liquid.
For measurement purposes it is convenient to estimate the acoustic impedance as p/v , which
from Equations (2) and (3) after application of the Fourier transform to the pressure and
velocity variables yields:

ˆ

p 



i  i  sin kL

ˆ

0 0

Z  tan kL 

(6)

a 0

ˆ ˆ ˆ

v k k cos kL

Where p  and v  are the Fourier transformed pressure and velocity respectively. The acoustic
impedance is a complex number, as well as its inverse, which is known as mobility. Thus,
the magnitude or absolute value of the mobility Abs(1/Z a0 ) can be calculated. From values of
the acoustic impedance or mobility the complex wave number k can be obtained as well as
its real and imaginary components, from which the viscosity of the liquid and the intrinsic
sound velocity in the liquid of interest can be estimated from the following equation
(Temkin, 1981):



0.5 0.5 0.5 0.5



 

2 2

 

i              

ˆ

 

kk    

    (7)

     

1

  

2 2

RC 2  2 C  3  4 C 



     

    

1



1



1



Viscosities of the liquids can be also estimated from the Kirchhoff's equation, which has
been derived for waveguides containing a gas and are given by the following equation
(Kinsler et al., 2000):

1 

 

(8)

RC 2 

1 0

The parameter R in Equations 7 and 8 is obtained from to the magnitude of the mechanical
impedance at the resonance frequency (see Figure 2).
From Equation (6) can be observed that plots of the acoustic impedance becomes a
maximum when

ˆ

cos( kL ) is a minimum, i.e. when the frequencies are given by the

C 1

1

relationship f  ( n  ) , where n = 0,1,2,3,…., which is in terms of the length of liquid L

2 L 2

and the velocity of sound in the liquid C 1 , can provide a location of those maxima. Those
theoretical observations were experimentally validated by Mert et al. (2004) using an
experimental setup as that described in Figure 1.