The right term inside the brackets represents the direct contribution of the vibration 

 to the radiated sound; if the displaced mass pV is of the same order as the cylinder 

 mass, the contribution can be considerable, especially since 8 is often smaller than 0.01 

 for ordinary mechanical systems. The vibration can thus cause a large increase in the 

 sound pressure in liquids, independently of any influence on the vortex shedding or 

 other aspects of the gross flow. The vibration will, in fact, modify the gross flow and 

 the associated fluctuating force. This interaction has not been studied in any detail. 



These aeolian sounds are associated with fluctuations at a predominant fre- 

 quency. However, the fluctuations usually contain a random component. In the flow 

 past a cylinder, the fluctuations become almost completely turbulent at high Reynolds 

 number, above about 10 5 . The radiated sound is then also random. The spectral dis- 

 tributions of the random force and vibration are related by an expression similar to 

 Eq. (28); if the vibration contributes to the sound, the sound spectrum will be peaked 

 around the resonant frequency of the system. 



The foregoing discussion indicates that there is a satisfactory understanding of 

 the sound generated by flow about a cylinder. However, this case is only the simplest 

 example of the acoustic interaction of a surface with an unsteady flow. A more compli- 

 cated situation involves the sound associated with a turbulent boundary layer. This 

 is of practical interest in connection with sound generated at the skin of an aircraft 

 or at the hull of a ship. 



Noise from a boundary layer. — The sound generated by the fluctuations in the 

 boundary layer on a rigid bounding surface have been investigated theoretically by 

 Phillips. [56] His results indicate that the sound from a plane boundary layer is small, 

 except perhaps near the transition region. The radiated sound pressure is proportional 

 to pU a 2 (U /c) if the flow maintains similarity. Accordingly, it is likely that only 

 negligible sound will be radiated at the speeds encountered in water. 



If the boundary surface is relatively flexible, however, flexural vibration of the 

 surface can result in significant sound. This surface vibration is excited by the local 

 pressure fluctuations within the boundary layer; the resulting sound pressure can be 

 much larger than that radiated by the pressure fluctuations themselves. The vibrating 

 boundary acts like a sounding board and substitutes a simple source with no (U /c) 

 dependence for the dipole-like source of a rigid boundary. 



A calculation of the radiated sound can be made with the following sequence 

 of steps: 



1. The pressure fluctuations at the boundary are estimated from a knowledge 

 of the flow. 



2. The flexural vibration of the boundary in response to the pressure fluctua- 

 tions is determined. 



3. The radiated sound pressure associated with the vibrating surface is calculated. 

 This procedure assumes that the motion resulting from the vibration is too small 



to modify the grosser flow, so that the pressure fluctuations at the boundary can be 

 treated as an independent variable. 



Once the boundary pressure is known, steps (2) and (3) involve the application 

 of well-known acoustical equations which give the sound pressure radiated by a mem- 

 brane or a plate in terms of an arbitrary distribution of pressure on its surface. Because 

 the fluctuations in the boundary layer are random in both time and space, the calcula- 

 tions are relatively complicated, but no essential difficulty is introduced by the random- 

 ness. It is necessary to know the statistics of the fluctuations, viz., the special density 

 and the space correlation of the pressure fluctuations. The evaluation of these quanti- 

 ties, however, which is essentially the first step listed above, has not been accomplished 

 in a completely satisfactory manner as of this writing. A theoretical estimate of the 

 characteristics of the pressure fluctuations can be attempted by following the procedure 

 outlined by Batchelor and already discussed in connection with the local fluctuations 

 within a turbulent region. However, too little is known about the fluctuations within 

 the boundary layer to permit the calculation to be carried out with any rigour. 



272 



