Figure 3 shows Q rad and Q vcs , 



increases with increasing co a and decreases with 



as functions of co a. Q VIS 

 increasing E,/a. Values of 



2 x 10 2 poise/cm < E,/a < 10 3 poise/cm 

 give values of Q V1S which are comparable to values obtained for Q in near-surface swimbladder resonance 

 experiments [15, 33, 36-38]. Q rad decreases with increasing a) a and, depending upon the value of E/a, may 

 be significant for all co a considered. 



Figure 4 shows Q th as a function of u) a, a, s/a, and depth D. Any individual numbered curve indicates 

 that Q th increases gradually with increasing s/a. The set of numbered curves indicates that Q th increases 

 with increasing a and D. The lettered curves are for the case where the effect of surface tension is 

 insignificant, and also indicate that Q th increases with increasing a and D. A comparison of figures 3 and 4 

 shows that Q, h is significant only at depths above 100 m and then only if 1,1a < 10 3 poise/cm. 



At off-resonance frequencies 



m a- • 



2s 



H„ 



(IV-7) 

 (IV-8) 



and 



H, 



(*)' 



1 + 



Po,ooo 2 a 3 



1 



2s 



p 0f o)2a3 



Qth 



(IV-9) 



Thus, at frequencies above resonance, H rad may be the dominant damping term, even for relatively high 

 values of E,/a. Conversely, at frequencies below resonance, H rad may be the least important damping term. 

 The variation of H th with (go /u)) is complicated by the presence of the surface tension term. It can be shown, 

 utilizing equation IV-1, that 



2s „ . _ (IV-10) 



1 < 1 + 



Po,<Va 3 



< 1.31 



so that, for the purpose of establishing a trend, 



H t h 



» (*)"( 



1 + 



2s 



P 0( 0)2 33 



Q„ 



(IV-1 1) 



Hence, at frequencies above resonance and at frequencies below resonance for which p 0f o) 2 a-»(2s/a), 



H th z {%y a* . 



At frequencies below resonance for which (2s/a),>> p 0f u) 2 a 2 , 



Thus, H th becomes relatively less important than H rad as frequency increases and conversely. 



(IV-12) 



(IV-13) 



Comparison to Free Bubble 



The validity of the results developed for the new model must be determined. This will be done by first 

 checking that the equations developed for the new model approach the equations for a free bubble in water 

 as a limiting case and then comparing values calculated using the new model with experimental values. 



The equations obtained in Chapter III for the resonant frequency and scattering cross section of the new 

 model can be readily compared to the equations given for a free bubble. Equations I-6 and I-7 give co and o 

 for an ideal bubble, neglecting surface tension. Equation III-26 reduces exactly to equation I-6 when E, = s = 

 0. If thermal losses are also neglected, equation III-33 reduces to equation I-7 when E, = s = 0, except for 

 factors of (p^/Po,). If surface tension is included, go for an ideal bubble is given by equation 1-12 with 

 n Sw = 0. Equation III-26 reduces exactly to equation 1-12 when E, = r) Sw = 0. Thus, when viscous and thermal 

 dissipation effects are eliminated from the new model, the results are equivalent to results obtained for an 

 ideal bubble. 



When the viscosity of water is considered for an air bubble in water, o> is given by equation 1-12. A 

 comparison of equation III-26 with 1-12 shows that the form of the viscosity factor for the new model differs 

 from that given for a bubble. However, as was noted earlier, equation 1-12 is only valid for small viscosities, 

 so that this difference is not surprising [25]. Both equations 1-12 and III-26 indicate that u) decreases as the 

 viscosity increases, which should be the case for a viscously damped system [51]. 



25 



