13:3/ Mechanical Resonances of Biological Cells 241 



The tangential-type modes are not affected by the intra- and extra- 

 cellular liquids, to the extent that these liquids may be considered as 

 having negligible viscosity. This tangential-motion-only mode may be 

 described by a displacement in the ifj direction only, which will be 

 denoted by *F. Because the liquids slip freely over the surface, these 

 modes and frequencies are independent of h. They are described by 



T = ^^oW^(cos^)],-^ 



(13) 



ag --£-.(»- l)(» + 2) 

 Ps a 



where A n is a constant, \x is the shear modulus and p s is the shell density. 

 Values of \x in the range of 10 3 dynes/cm 2 lead to the resonant frequencies 

 in the ranges observed for the optima for cellular destruction in cavitating 

 acoustic fields. 



In modes with both radial and tangential motion, there are both a 

 radial displacement R(r, 6) and a tangential displacement 0(r, 6). 

 (This argument could be made more general by including a displace- 

 ment *F, and allowing R, 0, and *¥ to depend on iff as well as r and 9. 

 However, very little is gained at the expense of making the notation 

 much more complex.) The problem with both R and cannot be 

 solved in a simple closed form analogous to Equations 10 and 13. How- 

 ever, the differential equation can be satisfied by 



R n = A n P n (cos6)e- i » n t 

 and (14) 



=A *£ 



n n dd 



where A n is a constant and A n a function of a, h, /x, and Poisson's ratio. 



For given values of these parameters one can also find three values for 

 a> n . Two of these lead to absurd numerical contradictions. The third 

 value of a> n is in the range observed for optimum cellular destruction, 

 if the shear modulus /x is around 10 3 dynes/cm 2 , and hfa is in the range 

 of 0.1-0.2. 



Thus, this model predicts two different types of modes in the observed 

 frequency range for cells having outer layers of protoplasm similar to a 

 protein gel. If detailed observations on the cell shape during such 

 resonance were possible, one could distinguish these two types of modes 

 from each other and from interfacial-tension modes. In the absence of 

 such observations, it is impossible to choose between these alternatives. 

 For example, a photograph of a red blood cell in a cavitating ultrasonic 

 field is shown in Figure 3. It strongly supports the existence of surface 



