E. J. Skudrzyk and G. P. Haddle 259 
and 2,2 6 1\ |] /27u.\n—-3 
P(w) = 1.5-107§ = Fe) -2)(2e for w > @o 
Measurements with a small receiver lead to a value m=3 for the slope of 
the curves in the descending part of the spectrum. For m= 3 the expression in 
square brackets reduces to 1, and the power spectrum of the flow noise at low 
frequencies becomes proportional to the third power of the velocity and to the 
boundary-layer thickness; and proportional to about the sixth power of the ve- 
locity and inversely proportional to the square of the boundary-layer thickness 
at high frequencies. These predictions and the numerical values computed by 
the foregoing equation are in very goodagreement with the experimental results, 
as will be shown later. 
14.3. THE RADIATED FLOW NOISE 
Lighthill's theory [13] leads to the computation of the farfield sound pres- 
sure. Unfortunately, the magnitude of this pressure can only be given in terms 
of higher order correlations, which have not yet been studied in detail. It has, 
therefore, not yet been possible to derive a correspondingly simple formula for 
radiation-field sound pressure generated by the turbulent velocity fluctuations. 
It will subsequently be shown that the sensitivity of a hydrophone to the flow 
noise nearfield decreases greatly with the diameter of the hydrophone. A large 
hydrophone is, therefore, practically insensitive to the flow noise nearfield. But 
a large hydrophone is sensitive to the radiation-field pressure that is generated 
by the unsteadiness and the decay of the turbulence. This sound field propagates 
inside and outside the boundary layer and is correlated over distances of about 
half a sound wavelength. Since the hydrophone is usually much smaller than the 
sound wavelength, it is fully sensitive to the radiated sound and records it as 
flow noise. A large hydrophone measures almost exclusively the true sound that 
is produced by the unsteadiness of the turbulence. 
14.4. EFFECT OF SIZE AND SHAPE OF SOUND RECEIVER ON AMPLITUDE-VS-FREQUENCY 
CURVE OF FLOW NOISE 
Flow-noise pressure is a local, rapidly varying quantity and only a hydro- 
phone that is small in comparison to the scale of the local pressure fluctuation 
measures the true value of pressure generated by the turbulent velocity fluctua - 
tions. If this hydrophone is tuned to a narrow frequency band, its response 
becomes proportional to the Fourier component of the noise pressure in the 
received frequency band. It has beenshownby Taylor, Proudman [14], and others 
that the decay of the turbulence has only a minor effect on the nearfield noise 
pressure. It can be assumed that the turbulence is carried along by the flow, 
with a mean velocity UY not very different from the free-stream velocity of the 
flow. The frequency f of noise pressure received by the small hydrophone, and 
the space wave number x of the turbulence are therefore connected by the 
relation, 
Pal, aH 2b i ako (12) 
