Frequency in cycles per second 

 100 1000 



10000 

 +30 



001 



0.1 I 



Strouhal Number f D/ U 



Figure 20. The spectral density of the fluctuating pressure in the near-field of a free jet of water. 



(From Jorgensen [21].) 



flow in the presence of boundaries were developed by Curie [52] in 1955. If the 

 boundaries are directly within the unsteady region, their principle contribution to the 

 radiated sound pressure is given by a surface integral over the boundaries, of the pres- 

 sure and the boundary acceleration, viz., 



p fu/ i a />' 



Vs{x,t) = / — cos 6jdS H / — cos dj dS. (24) 



4tJ r 4x dxj J r 



Here r is the distance between the point x and the surface element dS; p f is the pres- 

 sure and u v ' the component of acceleration in the x- t direction, at the surface element 

 dS at earlier time t-r/c; and di is the angle between the x. h direction and the surface 

 normal out of the fluid. It is of interest that the equation is of the same form as the 

 classical solution of the homogeneous wave equation in terms of the pressure and 

 acceleration at the boundary (cf. Lamb [6] §290); the same terms appear in Curie's 

 result even when the motion near the boundaries is so large that the homogeneous wave 

 equation is no longer applicable. 



If the boundary is contained within a region small compared with the wave- 

 length of the sound at the frequency of interest, the integrals in Eq. (24) may be 

 evaluated without reference to the shape of the boundary. In the radiation field far 

 from the boundary, the sound pressure is then given by the simple relation 



1 



Ps(t) = 



4ttV 



P VU F 



pV H cos 6 u -\ cos 6 F 



c c 



(25) 



269 



