where V is the instantaneous volume enclosed by the boundary; U the velocity of the 

 centroid of the enclosed volume; F the total force exerted on the fluid by the boundary; 

 and U and d P the angles between the radius vector toward x, and the directions of 



U and F respectively. The dots indicate time derivatives. In this equation, the terms 



on the right have their values at earlier time (t-r/c). The V term in the equation 

 corresponds to the ordinary simple source associated with volume pulsation (cf. Eq. 

 (4) of Section 2). The other two terms have a dipole-like directionality and are asso- 

 ciated with translational oscillation of the boundary and with oscillating forces on the 

 fluid.* 



Even if the boundary is rigid and does not vibrate, sound can nevertheless result 

 from oscillating forces acting on the fluid. Thus, the aeolian tones radiated by a cylin- 

 der have been explained in terms of the oscillating force, between the fluid and the 

 cylinder, associated with periodic shedding of vortices. To calculate the magnitude of 

 the sound, Etkin, Korbacher and Keefe [53] assume that the transverse oscillating force 

 on the fluid is given by 



F = 0.90 (% P U 2 )LD sin (2tt/iZ), (26) 



where U is the free-stream velocity, D and L are here the diameter and length of the 

 cylinder, and f x is the frequency of vortex shedding given by j A — 0.2U /D. Substi- 

 tuting Eq. (26) into (25), and assuming that the cylinder is rigid, the sound pressure 

 is calculated as 



p s (t) = 0.18(y 2 pU 2 )(Uo/c)(L/r)(cos 2rfa) cos 6, (27) 



9 being the angle between the radius vector and the flow direction. The equation indi- 

 cates that the sound pressure is proportional to the cube of the velocity."* This relation 

 was verified by their experimental data, some of which are shown in Fig. 21. 



Similar calculations and measurements are reported by Phillips. [54] He calcu- 

 lates the fluctuating force from measured values of the fluctuating velocity in the wake 

 of the cylinder and obtains a coefficient 0.76 instead of the 0.9 in Eq. (26). The 

 measured sound pressures of both Etkin et al and Phillips are lower than the value 

 given by Eq. (27) by constant factors. The difference is explained by the fact that the 

 vortex shedding is not correlated along the entire length of the cylinder, so that the 

 phase of the oscillating force varies axially. If the correlation length is assumed to 

 equal the cylinder diameter times a factor b, then the quantity (bDL) 1 / 2 should be 

 substituted for L in Eq. (27). Phillips estimates that b is about 17 for the range of 

 Reynolds number U D/ V from 80 to 160, and about 3 for Reynolds numbers above 

 300; whereas Etkins estimates b as about 8. 



The intensity of aeolian sounds has also been measured by Gerrard [55] over 

 a wide range of Reynolds numbers. His data can be interpreted as indicating that the 

 sound pressure varies with the cube of velocity, as required by Eq. (27), but that the 

 numerical constant is larger by a factor of about 4 for Reynolds numbers below 300. 

 This higher value may be due to the fact that, at low Reynolds numbers, the vortex 

 shedding is correlated all along the cylinder axis. 



Both Phillips and Etkin et al assume that the vibration of the cylinder does not 

 radiate any significant sound. The latter, in fact, report measurements showing that 

 the sound is independent of the elastic properties of the cylinder. However, it is well 

 known that the intensity of the aeolian tones increases when the cylinder vibrates in 

 resonance at the shedding frequency; this phenomenon was observed by Strouhal him- 

 self. Phillips' explanation is that the vibration of the cylinder "locks-in" or correlates 



* High-order terms, with higher negative powers of c, have been omitted from Eq.(25); 

 such terms are associated with higher moments of the boundary vibration and pressure dis- 

 tribution. 



** Eq. (27), without the numerical coefficient, was predicted by Blokhintzev, ibid. 



270 



