242 Mechanical Resonances of Biological Cells /1 3 : 4 



modes of resonance, but it is impossible to determine the number of 

 nodal diameters, much less examine the details of the shape necessary 

 to distinguish various modes. 



Figure 3. Photographs of rat red blood cells in ultrasonic 

 fields. Note that there are some undistorted cells and some 

 showing various modes of distortion. After L. Binstock and 

 E. Ackerman. 



4. More Exact Treatments 



In deriving the expressions for the resonant frequencies in Sections 2 

 and 3, a number of assumptions were made. The validity of these is 

 considered in more detail in this section. To a physicist, probably the 

 most noticeable assumption was the absence of viscosity in the fluids. 

 Readers with more biological training would emphasize the nonspherical 

 shape of all real cells. Other factors, explicitly or implicitly neglected, 

 include the effects of the compressibility of the liquids, the relationship 

 of breakdown rate to resonance, and the actual modes present. 



The major effect of viscosity is to damp any free vibration. With the 

 geometries chosen in this chapter and typical viscosities measured for 

 protoplasm, that is, coefficients of viscosity from 2 to 10 centipoise, this 

 damping is very pronounced. The net result is to broaden the resonance 

 curve as shown in Figure 4 ; the curve including viscosity has a mechani- 

 cal Q of the order of 2. 1 Nonetheless, the resonant frequencies are only 



1 The quality factor Q of a resonance may be defined in several equivalent 

 forms. For the purpose of this chapter, it may be considered to be defined by 

 the relationship 



Q =/o/A/ 



where A/ is the width of the band between the two frequencies at which the 

 square of the amplitude of the response of a vibrator is decreased by a factor of 

 two from its value at the resonant frequency f provided the vibrator is driven 

 by a force of constant amplitude. The greater the damping, the broader will be 

 the resonance and the lower the Q. 



