260 Lecture 14 
where A,, is the space wavelength of the pressure pattern that produces the noise 
band. Thus, the pressure fluctuations that give rise to a definite spectral com- 
ponent of the flow noise are no longer uncorrelated but correspond to a sinusoidal 
pressure pattern p = p(k) cos k(x-Ut) that moves in the direction of the flow. Thus, 
tuning the hydrophone destroys the randomness of the received signal. 
A hydrophone of finite size measures the average pressure over its area. 
If the hydrophone is infinitely narrow, but of a length 1,, this average pressure 
fluctuation is given by the integral 
1,/2 
F@)_1 p(k) cos x(é — Ut) dé = PL) sink aN sink Dh, ut 
ly ly ; Kit 2 2 
1/2 
— 2p(k) <3, Ka _ p(k) sin («1,/2) 
= ae sin 5 COS <$—arigy (13) 
where w=xU. The average value of spectral amplitude of the noise pressure over 
the hydrophone area is thus proportional to the infinite space spectrum p(x) of 
the pressure fluctuations in the boundary layer; p(x) alone determines the fre- 
quency component F(@) where w = Uk, whether the hydrophone is finite or infinitely 
small. The effect of the finite dimension of the hydrophone in the direction of 
the flow on the received pressure amplitude is represented by the factor 
sin(«l,/2) ~. [«l,\7? 
GA F é) (14) 
The maxima and minima in the pressure distribution cancel one another over 
the hydrophone area and the contribution of one pressure maximum, on the 
average, remains, irrespective of the length of the hydrophone; the contribution 
of this maximum is represented by the above formula. The longitudinal dimen- 
sions of the hydrophone become important as soon as the length of the hydro- 
phone is greater than one third the space wavelength 4,, of the turbulence that 
generates the noise. Doubling the length of a tuned hydrophone should then re- 
duce its relative flow-noise output by 6 db. This cancellation of successive 
maxima and minima introduces the factor 1/x? =(U/a)? in the frequency curve of 
the measured flow-noise power spectrum. 
If the hydrophone is of variable width, but of a width that is always less than 
the transverse correlation length of the pressure fluctuations, the integrand 
must be multiplied by the width. If the hydrophone is symmetrical with respect 
to its midpoint, the mean force on its membrane becomes 
U 
FO) - f eee =U) 1 ce = (a5) 
0 
The mathematics, then, is exactly the same as that for a shaded hydrophone 
array, the magnitude «/sin 6 being replaced by «/, and the shading function being 
