Underwater sounds are ordinarily detected by devices sensitive to pressure 

 fluctuations. These devices are called "hydrophones," and they are simply waterproofed 

 microphones. Accordingly, the physical quantity of interest in hydrodynamic noise is 

 generally the fluctuating pressure associated with the hydrodynamic flow. The magni- 

 tudes of the fluctuations of interest are very small compared with the pressures 

 ordinarily met in steady hydrodynamics: a pressure fluctuation of only 100 dyne/cm 2 

 (a head of about 0.04 inch of water) is an intense sound in water. The power associ- 

 ated with these fluctuations is also very small: the entire underwater sound output of 

 a noisy ship is only a few watts. It is thus apparent that hydrodynamic noise may 

 stem from second-order effects which have no influence at all on the more obvious 

 characteristics of the flow. 



It is convenient to classify the various forms of hydrodynamic noise according 

 to the grosser flow phenomena with which they are associated. Four categories will 

 be considered here: 1. Air bubbles entrained in water, 2. Vaporous cavitation, 3. Sur- 

 face disturbances such as splashes, and 4. Unsteady flows such as vortex shedding and 

 turbulence. Each of these phenomena requires somewhat different methods of analysis; 

 accordingly they will be discussed in separate sections, the order being chosen merely 

 for convenience of exposition. 



II. Entrained Gas Bubbles 



Perhaps the most commonplace of all the sounds in water are those associated 

 with bubbles. Sound is generated when bubbles form, when they divide or unite, and 

 when they stream past an obstacle in a flow or through a constriction in a pipe. 



The sound of bubbles forming at a nozzle was investigated by Minnaert [3] in 

 1933, and later by Meyer and Tamm [4], who showed that the sound was associated 

 with volume pulsations of the bubble, and that in these pulsations the bubble behaved 

 like a simple oscillating system with damping. The frequency of the pulsation was 

 calculated by Minnaert and the damping by Pfriem [5] among others. 



When bubbles are observed as they rise through a liquid, large oscillations in 

 their shape are apparent. It is natural to ask whether the sound associated with these 

 shape changes is significant relative to the sound from the volume pulsation. This 

 question can be answered by representing the vibration of the bubble wall by a sum 

 of spherical surface harmonics (see Lamb [6] §294) 



#(0,<M) = #o + 2 n A n S n (d,<t>)eWJ, (1) 



where R is the instantaneous radial coordinate of a point on the bubble wall, as a func- 

 tion of the spherical angles and <p, and time t; R is the mean radius of the bubble; 

 S n (0,(f>) is the surface harmonic of order n giving the variation with the angles; 

 i ~ V — 1; and A n and f n are the amplitude and frequency of oscillation for the n-th 

 order. The zeroth order, with S — 1. corresponds to volume pulsation, the first order 

 to translational oscillation, and the higher orders to oscillation in shape with n nodal 

 lines. 



If the amplitude of oscillation is small, i.e., if A n <^.R , the oscillations of each 

 mode contribute independently to the sound pressure. Far from the bubble, the sound 

 pressure amplitude p n associated with oscillation in the n-th mode at frequency / is 

 given by 



P A n R i (27rfye- ikr (ikR ) n S n 



Pn = , (2) 



r(n + 1) [1-3-5 • • • (2n - 1)] 



where r is distance from the bubble, and k — 2-rrf/c. The derivation of this equation 

 assumes that the bubble is small, so that kR <^l, and that the distance is large so that 



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