The damping in the new model can also be compared to that for a real bubble in water as determined by 

 Devin [19]. For ease in comparison with Devin's results, equations IV-4, IV-5, and IV-6 will be rewritten as: 



5 rad = ££^£i , (IV-14) 



5 * s = dd^p • < IV - 15 > 



and 



5 th = *"' ~ V(- K * V fl + "\ . (IV-16) 



Wo^a \2p 0a c Pa ) \ Po f co 2 a3 ) 



Equation IV-14 is equivalent to equation I-9, except for the factor (p^/Po,). Devin has considered only the 

 shear viscosity, so that equation 11-1 appears to indicate that equations IV- 15 and 1-10 differ by a factor of 

 %. However, n, b and n, s are related to the dilatational viscosity r) d , by [45] 



H b = rid + %ns ■ (IV-17) 



Therefore, if r\ is neglected, equation 11-1 indicates that E, = 2n. s - Hence, equations IV-15 and 1-10 are 

 equivalent. Equations IV-16 and 1-1 1 are also equivalent, except for the surface tension factor in equation 

 IV-16. Devin considered surface tension, but since the surface tension of an air bubble in water is only 74 

 dynes/cm, its effect is insignificant for the frequencies and bubble sizes of present interest. Hence, the form 

 of the damping factor at resonance for the new model is essentially equivalent to that of a real bubble in 

 water. 



Comparison to Experimental Data 



The previous comparisons compared the spherical model to spherical bubbles. In order to compare the 

 model to experimental measurements made on swimbladder-bearing fish, it is obvious from figure 1 that the 

 effect of swimbladder shape should be included. This is done by letting 



co 0c = <> 0t , (IV-18) 



where co 0t is the theoretical resonant frequency for the spherical model, £ is given by equation I-25, and o) 0c 

 is the value which is utilized in the comparisons to the experimental data. 



There are five sets of experimental data to which calculations based on the new model can be 

 compared. These are contained in references 15, 33, 36, 37, and 38. In each of these, the measured 

 resonant frequency, co 0m , and the calculated Q are either given directly or can be obtained from a curve. The 

 data in reference 36, which was collected a decade before any of the other data, have variations which 

 require some interpretation to obtain CLb m and Q. This required interpretation makes the a> 0m and Q values 

 obtained from reference 36 less reliable than those obtained from other sources. 



Comparisons of the data to the model require that values for a and e be known. In some cases, a and e 

 were measured and the values given. In the other cases, it was possible to determine a indirectly from other 

 data in the report. When values of e were not given, average values were obtained from other sources. 

 Measurements of a and s were all made at atmospheric pressure. There is some question as to what effect 

 increasing depth has on swimbladder volume [e.g. 11, 12]; that is, whether the fish retains a constant 

 swimbladder volume or a constant swimbladder mass (Boyle's Law) or some other, intermediate process. 

 (McCartney and Stubbs [37] assumed an intermediate process, due to the effect of tension in the 

 swimbladder wall.) It is probable that different species react to changing depth in different ways and that 

 different experimental methods can affect the way a fish would normally react. Hence, there are 

 uncertainties in the actual values of a at depth. Since the swimbladder is attached to various other parts of 

 the fish, there is no reason to expect a change in swimbladder volume to cause a uniform change in its 

 linear dimensions. Thus, the uncertainty in a at depth produces a smaller uncertainty in e. Various depth 

 variations in a were examined when comparing the data to the model. However, since the individual 

 researcher is in the best position to interpret his own data, the final comparisons utilized the depth variations 

 chosen by those researchers. 



The new model requires values of E, and s. Since the exact value, or range of values, of ^ is presently 

 unknown and s is quite likely under the control of the individual fish, direct comparisons of the model and 



28 



