^7 205 



Then, finally, there are the wavelets scattered In all directions 

 by the bubbles. In part, effects of these scattered wavelets have already 

 been taken into account, for they actually constitute the physical mechanism 

 by which the incident wave is weakened and partly reflected. But the scat- 

 tered wavelets will also appear Independently as an additional wave of pres- 

 sure scattered in all directions. In a similar way the waves of light 

 scattered by the molecules of the atmosphere, which, on the one hand, cause 

 a refraction and a weakening of the sun's rays, also appear independently as 

 the blue light that comes from the sky. Scattered wavelets coming from more 

 and more distant parts of the bubbly layer may prolong the transmitted wave 

 as observed in regions beyond the layer. 



The momentum carried by the waves, on the other hand, should be re- 

 duced only if a reflected wave of tension occurs. Such a reflected wave may 

 carry back a large part of the incident momentum. If, however, the reflec- 

 tion is prevented by the occurrence of cavitation, all of the incident mo- 

 mentum must appear somehow in the transmitted wave. 



The transmitted momentum might, as a matter of fact, exceed the in- 

 cident momentum. In such a case the conservation of momentum might be pre- 

 served in either of two ways. Partial reflection from the farther side of 

 the bubbly layer, occurring in the medium of lesser acoustic impedance, may 

 cause momentum reversed In direction to be carried back toward the source of 

 the waves. Or, momentum of the same sort may be carried back by the rear 

 halves of wavelets scattered off in all directions. 



Several features corresponding to those Just described have been 

 observed in experiments at the David W. Taylor Model Basin, which are to be 

 described in other reports. 



The quantitative treatment of these phenomena, unfortunately, en- 

 counters great difficulty, as does any problem in highly non-linear wave 

 motion. The analysis can be effected readily, in fact, only for the extreme- 

 ly simple case of very weak waves of sinusoidal form, passing through water 

 that contains many bubbles in each cubic wave length. After this case has 

 been solved, a weak wave of arbitrary form can be treated, if desired, by 

 means of Fourier analysis. Although devoid of direct bearing on the topic 

 of explosion waves, the analytical results for weak waves may be suggestive 

 enough to be quoted here. Their deduction is given in the Appendix. 



WEAK SINUSOIDAL WAVES IN FINE-GRAINED BUBBLY WATER 

 Let the following assumptions be made: 



1 . The pressure is so weak that linear acoustic theory can be applied. 

 This implies that the bubbles change size only slightly as the waves pass. 



