THE TESTING PROBLEM IN GENERAL 



35 



The characterization of a transducer lor practical 

 purposes is usually effected by evaluating the follow- 

 ing quantities: (1) receiving response as a hydro- 

 phone, (2) transmitting response as a transducer, (3) 

 directivity pattern, and (4) impedance. 



Receiving Response as a Hydrophone 



The receiving response as a hydrophone is defined 

 as the open-circuit voltage (in db vs 1 volt) generated 

 by the hydrophone when placed in a uniform plane- 

 wave sound field of reference pressure (1 dyne per sq 

 cm) propagating parallel to the acoustic axis of the 

 hydrophone. (See Chapter 1.) The acoustic axis of the 

 hydrophone is an arbitrarily selected axis through the 

 hydrophone, which is, however, usually chosen to be 

 either some axis of symmetry for the instrument, the 

 axis of maximum response, or some other readily 

 identified axis. The plane-wave sound field of 1 dyne 

 per sq cm should be, of course, the sound field when 

 the hydrophone is not present, since the latter will in 

 general distort the field by diffraction. If the hydro- 

 phone is linear, the voltage generated is proportional 

 to the magnitude of the sound field, and the measure- 

 ment may be made in a sound field of any magnitude 

 and then reduced to its value for a reference field. In 

 some cases, where it is not possible to measure the 

 open-circuit voltage of a hydrophone, the voltage 

 across some given impedance is measured. This volt- 

 age may or may not be reduced to an open-circuit 

 voltage, depending on the circumstances. 



Transmitting Response of a Transducer 



A transmitter of finite size can be shown theoreti- 

 cally to produce a sound field such that the pressure 

 along any axis of the transmitter, at sufficiently great 

 distances from the transmitter, falls off directly as the 

 distance from the transmitter. This region at suffi- 

 ciently great distances is known as the inverse-square- 

 law region, since the sound intensity falls off as the 

 square of the distance. Close to the transmitter the 

 sound field does not follow this law. Here a more 

 complicated sound field distribution obtains, which 

 depends upon the particular characteristics of the 

 transducer. For most applications only the pressure 

 produced in the inverse-square-law region is of im- 

 portance. 



The relationship between pressure and distance is 

 given by 



lor points in the inverse-square-law region, where p is 

 the pressure at a distance d from the transmitter on 

 any axis through the transducer, and C„ is a constant 

 which may have different values for different axes. 



The transmitting response of a transducer is de- 

 fined as the pressure measured at a distance d in the 

 inverse-square-law region on the acoustic axis of the 

 transducer for 1 watt available power from a given 

 generator impedance (assumed to be purely resistive), 

 reduced to a distance of 1 meter by multiplication by 

 d in meters, and expressed in db vs 1 dyne per scj cm. 

 (See Chapter 4.) The available power of a generator is 

 defined as the power delivered by the generator to an 

 impedance which is the complex conjugate of its own 

 impedance. Thus, if the generator has a purely resis- 

 tive impedance r g and a generator voltage e g , the 

 available power from the generator P A is given by 



Pa 



(#'• 



4r„ 



(2) 



P = C -i 



(1) 



(See Figure 1 in Chapter 4.) For a linear transducer 

 the generated pressure is proportional to the current 

 into the transducer (independent of its impedance). 

 Therefore, to characterize a transmitter on the basis 

 of available power, it is necessary to know not only 

 the impedance of the source but the impedance of the 

 transmitter as well. 



Directivity Pattern 



The transmitting and receiving responses charac- 

 terize the acoustic behavior of a transducer at long 

 distances on its acoustic axis; however, information is 

 also desired on the sound field produced in other 

 directions. This information is provided by directiv- 

 ity patterns, which give the ratio in db of the response 

 in any direction to the response on the axis. To char- 

 acterize directions, it is necessary to set up a spherical 

 coordinate system for the transducer. Referring to 

 Figure 1, we take the Z axis along the acoustic axis of 

 the transducer and choose the XZ plane in an arbi- 

 trary orientation which is usually selected to be the 

 horizontal plane through the transducer in its normal 

 operating position. The direction of any line can 

 then be specified by the angle 8 between the line and 

 the Z axis, and the angle $ between the XZ plane 

 and the plane through the Z axis and the line, as 

 shown. To specify completely the directional pattern, 

 the ratio of the response on every axis to that on the 

 acoustic axis should be given. If the sound field is 



