t and t' is taken into account in the evaluation of the integral, however, a term results 

 which is proportional to (p/rc 2 ). The pressure fluctuations associated with this term 

 vary inversely as the distance in the manner of ordinary spherical waves of sound. 

 Indeed, these fluctuations correspond to the sound energy radiated by the flow, and 

 this distant region is accordingly called the "radiation field." 



At intermediate distances, neither far away nor directly within the unsteady 

 region, the dependence or independence of the pressure fluctuations on the compressi- 

 bility of the fluid is determined by the frequency of the fluctuations. At low frequencies, 

 associated with sound of wavelength c/f much larger than the distance, the fluctuations 

 do not depend on the compressibility. When it is desired to distinguish the essentially- 

 incompressive pressure fluctuations existing within a wavelength of the unsteady region 

 from the compressive fluctuations in the radiation field, the close-up distances are called 

 the "near field." In the near field, the magnitude of the pressure fluctuations increases 

 more rapidly with decreasing distance than in the radiation field. 



Because the pressure fluctuations within and near the region of unsteady flow 

 do not involve the compressibility of the medium, these fluctuations have been called 

 "pseudosound" by Blokintzev. [41] However, a pressure-sensitive hydrophone responds 

 to the pseudosound just as it responds to any sound pressure; the fact that these fluctua- 

 tions do not involve propagated sound energy makes no difference. 



Fluctuations within the unsteady region. — An estimate of the magnitude of the 

 pressure fluctuations within an unsteady flow was made by Taylor [42] in 1936. He 

 related the pressure to the fluctuating velocity by an expression like Eq. (21) but with 

 the time-derivative term omitted; the omission is equivalent to the assumption that the 

 fluid is incompressible. To obtain a tractable velocity field which nevertheless dupli- 

 cated some of the characteristics of isotropic turbulence, the velocities were assumed 

 to be distributed in space like standing waves of sound in a box. For this synthetic 

 model of isotropic turbulence, the fluctuating pressure and velocity are related by 

 p — l.Gpu^, where u 1 is one component of the fluctuating velocity and the tilde (~ ) 

 indicates rms values of the fluctuations. 



More recently, calculations of the fluctuating pressure in isotropic turbulence 

 were made independently by Heisenberg, [43] by Obukhov, [44] and by Batchelor. [45] 

 Their calculations all involve equivalent assumptions concerning the statistical charac- 

 teristics of the distribution function for the velocities. These more fundamental calcula- 

 tions lead to the result p = OfipUj 2 , the coefficient being lower than Taylor's original 

 estimate. According to Uberoi, [46] however, the value of the numerical coefficient is 

 quite sensitive to the exact form of the statistical distribution function of the velocities. 



Batchelor also obtained a relation between the space correlation function of the 

 fluctuating pressure and the correlation function of the velocity. For the specific case 

 of a velocity correlation of the Heisenberg type, the calculated pressure correlation 

 falls to zero more rapidly than the velocity correlation, and the longitudinal integral 

 scale of the pressure is about half that of the velocity. 



Ogura and Miyakoda [47] used Batchelor's relations to calculate the spectral 

 density of the fluctuating pressure from simplified spectral functions for the fluctuating 

 velocity. These calculations indicate that the spectral density of the pressure falls more 

 rapidly with frequency, in the high-frequency region, than does the spectral density 

 of the velocity. 



The first attempt to measure the fluctuating pressure in an unsteady flow seems 

 to have been reported by Rouse. [48] Within a turbulent jet of air discharging into 

 free space, the fluctuating pressure and velocity were found to be related by p = 1.1 pu x ~. 

 Strasberg and Cooper [49] attempted some measurements of the fluctuating pressure 

 in the turbulent wake behind a cylinder. At a point 24 cylinder diameters downstream, 

 p = 1.7 ' pu{-. They also determined the spectral densities of the fluctuating pressure 

 and velocity. The spectra measured 24 diameters downstream are shown on Fig. 19, 



266 



