If P a >P , so that p e (t) is actually negative at t = 0, the bubble becomes statically 

 unstable and the non-linear time-derivative terms in Eq.(12) control the radial motion 

 of the bubble.** As p e {t) goes negative, the bubble grows to many times its equilib- 

 rium size and then, when p e (t) returns to positive values, the bubble suddenly collapses 

 and radiates a very large positive pulse (dotted curve), Noltingk and Neppiras [9] have 

 calculated the size to which the bubble grows and the magnitude of the pressure pulse 

 as a function of the ratio P a /P . 



During its growth, the bubble becomes filled primarily with vapor. The large 

 growth and subsequent rapid collapse of the bubble are characteristic of the behavior 

 of vaporous cavitation bubbles. Indeed, the transition from small to large amplitude 

 pulsations is just the transition to cavitation, with the bubble acting as a cavitation 

 nucleus. The sound associated with these large-amplitude pulsations will be discussed in 

 greater detail in the following section. 



III. Cavitation 



The origin of cavitation noise. — The earliest investigators of cavitation were 

 aware of the noise it makes. It was probably the loud hissing sound which first at- 

 tracted the attention of Osborne Reynolds [10] to the occurrence of cavitation in water 

 flowing through a constricted tube. He recognized the cause of the sound to be the 

 "boiling" of the water. During the First World War, it was known that cavitation of 

 ships' propellers radiated sound which could be heard underwater for great dis- 

 tances [11]. 



The sound is generated as a result of the growth and collapse of vapor cavities. 

 A cavity, beginning as a microscopic nucleus, grows when its environmental pressure 

 becomes sufficiently negative and collapses when the pressure is restored. Such be- 

 havior, with accompanying noise, is to be expected wherever nuclei in the water are 

 subjected to sufficiently extreme transient reductions in their environmental pressure. 



The sketch, Figure 3, depicts one common occurrence, the growth and col- 

 lapse of individual cavities in the pressure field produced by flow past a curved 

 boundary such as that of a propeller, a strut or similar appendage, or past a contrac- 

 tion in a conduit. The growth begins after a nucleus of some sort enters the region 

 of low pressure and the collapse takes place when the cavity is carried downstream 

 into a region of higher pressure. Various experimenters have employed high-speed 

 photography to demonstrate the existence of individual cavities, roughly spherical, 

 which grow and collapse in the manner indicated [12, 13, 14, 15]. 



Cavitation noise may be produced also in turbulent shear flows in pipes, jets, 

 or boundary layers; by water hammer and by acoustic radiation; and even as a con- 

 comitant of "steady" cavities. In each instance, the sound is related directly to the 

 kinematics of the individual transient cavities. The latter may appear as elongated 

 "cores" at the centers of vortices; in shear flows, they may be rather amorphous and 

 may undergo rapid distortion. Where the velocity gradients are mild, the cavities 

 tend to a spherical shape. Nearly all analytical treatments of the hydrodynamical 

 problems assume a spherical cavity. This is not completely unrealistic: the validity 

 of many of the relations derived is not seriously affected by departures from spherical 

 symmetry. 



Interest in the inception, growth, collapse, and rebound of vapor cavities stems 

 from various fields of interest: boiling of liquids, detergent and chemical effects of 

 ultrasonics, absorption of sound in water, efficiency of hydraulic machinery, and 

 pitting and corrosion of structural materials. A great amount of research has been 



** If the equilibrium size of the bubble is very small, the condition for instability is 

 not simply P„>P , but rather P a >P + (4V3 T/9R„) [1 + (PoR»/2T)V'i- If Pa exceeds 

 this value by an appreciable amount, the size to which the bubble grows before collapsing 

 is relatively independent of its original size R . 



247 



