13:4/ Mechanical Resonances of Biological Cells 



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slightly shifted. The inclusion of viscosity complicates the analysis but 

 has no effect on the orders of magnitude computed for the interfacial 

 tension T in Section 2, or the shear modulus \x in Section 3. It does, 

 however, show that all modes must depend on ip, the lowest possible 

 varying as P^cos #) e2iu ' • 



The effects of quite large departures from a strictly spherical shape 

 are much less than those due to viscosity. The exact shape is not 

 critical because neither the kinetic energy nor the potential energy 

 depends sharply on the shape. This independence of exact shape is 



Figure 4. Effect of the quality factor Q on the shape of the 

 resonance curve. 



common to many types of resonances in all phases of physics, not only in 

 elasticity. The exact shape of cells is important for such effects as fluid 

 flow past the cell and diffusion. However, the resonant frequencies are 

 very insensitive to changes in cell shapes. 



The effects of the compressibility of the medium are also very small. 

 This implies that surface modes are very hard to excite with plane 

 acoustic waves. In contrast, an extremely nonplanar waveform near 

 small centers of cavitation could easily excite surface resonances of 

 biological cells. Likewise, streaming near a solid-liquid interface could 

 excite surface resonances. 



The cell disruption versus frequency curves at acoustic pressure levels 

 near the threshold for cellular destruction can be characterized by an 

 apparent Q that may be as large as 6. This sharpness indicates that 

 close to the threshold for cellular rupture, the rate of destruction increases 

 much more rapidly than the amplitude of resonant vibration. In 

 contrast, at higher acoustic pressure levels, the apparent Q drops to 

 values predicted for the vibration amplitude versus frequency curves. 



Finally, it should be noted that the order of magnitude calculations in 



