CHOICE OF TESTING GEOMETRY 



53 



or its absolute value is 

 dp 



J 



m- 



(39) 



Where r is large and the wave front is essentially 

 plane, this ratio becomes simply k. Therefore, if a 

 pressure-gradient hydrophone, calibrated in terms 

 of the equivalent plane-wave pressure, is placed in 

 a spherical sound field at a distance d from its center 

 and with its axis radial, it then indicates a pressure 

 which is greater by the ratio of equation (39) to /»-, or 



times the actual pressure present at its location. Note 

 that d now replaces r in equation (39). Thus, the 

 hydrophone indications should be corrected by this 

 factor to obtain the true pressure. This correction 

 factor in db is plotted in Figure 1 3 for four values of 

 d. It is to be noted that it is most prominent at low 

 frequencies. The correction factor is to be subtracted 

 from the observed pressure in db to obtain the cor- 

 rect value. 



One might conclude from the above that one may 

 work at any testing distance with a pressure-gradient 

 instrument and simply apply the above correction. 

 However, it must be remembered that most trans- 

 ducers do not have a spherical wave field in their 

 immediate neighborhood. This is true in particular 

 for piston-like transducers, where the sound field in 

 the immediate neighborhood of the face of the trans- 

 ducer is very complicated and does not become 

 spherical for what is often a considerable distance 

 from the diaphragm. Consequently, great caution 

 must be used in applying this correction. 



5.4.4 Proximity Effect for Pistons: 



Axial Response 



Most acoustic transmitters and receivers are cou- 

 pled to the acoustic medium by a diaphragm which 

 oscillates in a direction normal to its plane under 

 the influence of the pressure in the sound field when 

 receiving, or under the electromechanical forces of a 

 transducer when transmitting. The sound field of 

 such a piston source of finite area falls off according 

 to the inverse-square law at large distances, where 

 the wave fronts are spherical. Close to the transducer, 



0.1 



KC 



10.0 



Figure 13. Increase in sensitivity ot pressure-gradient 

 hydrophone in a spherical sound field. 



however, the sound field is very complex and cannot 

 be considered spherical. Only the case of a piston 

 situated in an infinite rigid baffle is amenable to 

 simple theoretical analysis because then there are 

 no edges to cause diffraction. The present discussion 

 is limited to this case. The results may be applied 

 satisfactorily to any piston whose dimensions are not 

 small compared to a wave length. 



Consider first the case of a circular piston. The 

 pressure along a line normal to the plane of the circle 

 at its center is, at a distance r from the piston. 



= 2 P cv, 



|sh4(vWr2-r)|, 



(41) 



where p is the density of the medium, c is the sound 

 velocity in the medium, v is the normal velocity of 

 the piston, /; = 2ir/\, and a is the radius of the pis- 

 ton. 78 This obviously does not vary inversely with r 

 for small values of r. However, as r becomes large 

 compared to a, the radical may be expanded to obtain 



P = 2pCV t> 



sm 



kjP 

 ir 



(42) 



If — — = —'— << I, the sine function may be expanded 



4r 2\r ' ' 



to obtain 



_ TTd-pCV, 



(43) 



where p„ is used to indicate the pressure at distances 

 where 



r >> — and ^>> 1. 



A a 2 



(44) 



In this region, the inverse-square law does hold, and 

 the pressure does vary inversely with r. 



