208 



20 



0.8 

 K 

 06 



Figure 7 - Reflection Coefficient K 



K is the fraction of the incident energy that 



is reflected, for the bubbly water 



to wMch Figure 6 refers. 



c/c' , or refractive index of bubbly 

 water relative to homogeneous water, 

 .and the quantity p are plotted 

 against w/w^, which represents the 

 ratio of the wave frequency to the 

 natural frequency of the bubbles. In 

 Figure 7 is plotted the reflection 

 coefficient K of the bubbly water. 

 The curves are valid for any bubble 

 size that is not too large; the size 

 of the bubbles determines w^, 



A strong resonance effect 

 is brought into evidence by these 

 curves. Especially striking is the 

 persistence of this effect as w in- 

 creases above w^. The decrease in 

 wave speed c that is caused by the 

 bubbles at low frequencies is re- 

 placed, as u begins appreciably to 

 exceed w^, by an increase in wave speed; for / = 0.001 and u = 2.'yu)^, c' = 

 22 c, and even at w = 3<^o> C = 1.7 c. Furthermore, the scattering, which is 

 proportional to yS, shows a strong persistence at values of u/uq up to 2 or 3. 

 As a consequence, there is a strong band of nearly total reflection from 

 (i) = ojg to cj = 3W(,. Above w = 5<^o' °^ the other hand, reflection becomes in- 

 appreciable; at such frequencies, the inertia of the bubbles prevents them 

 from following the vibrations of the incident wave to any considerable degree. 



If w is increased to very high values, however, a point is ulti- 

 mately reached at which the assumptions underlying the analysis no longer 

 apply, because the incident wave length is no longer large as compared with 

 the spacing of the bubbles. 



Observations have been reported on the scattering of sound by bub- 

 bly water, but they do not seem to lend themselves to a test of these equa- 

 tions. The analytical results may be employed, however, to throw some light 

 upon the effect to be expected when a shock wave enters bubbly water. A 

 photograph of this phenomenon is shovm in Figure 8. 



If the effective length of the shock wave is relatively great, or 

 at least not less than a third as great as the wave length corresponding to 

 the average nai^ural frequency of the bubbles (u/uq < 3), then it may be con- 

 cluded with safety that the reflection will exceed the value given by 



