48 



TESTING TECHNIQUE 



out effectively. It is also equal to /„/ A/ , where A/ is 

 the difference between the two frequencies, one on 

 each side of resonance, where the current falls to 

 l/\/2 times its value at resonance. Thus, one may 

 consider it a measure of how peaked the response 

 curve is and, therefore, a measure of how rapidly the 

 current response varies with frequency. For more 

 complicated circuits one cannot define an unambigu- 

 ous Q, but may often speak of an effective Q, valid for 

 some frequency range, such that Q measures the 

 number of cycles required for the transient to die out. 

 Associated with it is an effective time constant for the 

 circuit t which is approximately related to Q by the 

 relation Q = ttJ„t. 



A transducer has many similarities to an electric 

 circuit, and its response to a suddenly impressed 

 signal can be judged by an effective time constant, or 

 Q, for the transducer. In order to measure the steady- 

 state response of a transducer, one must therefore 

 wait a time considerably greater than t alter the 

 initiation of the signal, say t,„. Its value must be at 

 least that expressed by 



> ITT. 



(24) 



To delineate carefully the response peak of a reso- 

 nant transducer, one must have a resolving power 

 greater than the ratio of resonant frequency to the 

 breadth of the resonance peak, that is 





*7i>i 



(25) 



Thus, we see that if we set the resolving power equal 

 to /of,,,, as we conjectured earlier should be the case, 

 we again get the condition t,„ > 7tt, which is in 

 agreement with the result obtained from the con- 

 sideration of time constants. 



It then follows from the condition 



RP< 



fAL 



(26) 



that the resolving power is limited in the same way as 

 for warbled frequency or noise band signals, thus 

 showing that pulses are likewise ineffective at low 

 frequencies. A general criterion for the usefulness of 

 pulses in obtaining the response of a resonant trans- 

 ducer is 



(RP)™* = ^^ >Q- = 4t < 27 > 



C A J, j 



AL > 



A/o 



Qc 



h' 



(28) 



The problem of determining the time constant of 

 a transducer is not an elementary one in itself. In 

 many cases one can determine it a posteriori for a 

 resonant device, using the resultant Q obtained from 

 the response curve by a pulse method. If this (J is not 

 less than 7r/,,T„,, there will be considerable doubt 

 that the resolving power is greater than the Q. By 

 viewing the received pulses of the transducer on a 

 cathode-ray oscilloscope, one can usually obtain a fail- 

 estimate of the time constant. It should be pointed 

 out that a transducer, in contrast to an electric cir- 

 cuit, has its time constant determined not only by its 

 electrical and mechanical elements but also by its 

 acoustic geometry. In order for the local sound field 

 surrounding the transducer to build up to its steady- 

 state value, one must allow sufficient time for the 

 sound waves to pass from one part of the transducer 

 to another and build up the characteristic diffraction 

 pattern about the transducer. The time required for 

 the sound field to reach its local steady-state value is 

 of the order of the linear dimensions of the trans- 

 ducer divided by the velocity of sound. For a long-line 

 hydrophone and even for smaller instruments, the 

 acoustic time constant may be longer than that due to 

 the electrical and mechanical elements. In any case, 

 no response measurement should be made before the 

 acoustic pulse has enveloped the transducer. 



The pulsing technique is particularly valuable in 

 measuring directivity patterns, where the reflected 

 signals may be higher in level than the direct signal. 

 This is a case where other available methods often 

 fail (see conclusion of Section 5.3.5). 



Methods for producing and measuring pulses are 

 discussed in detail in Chapter 6 along with practical 

 considerations in employing the pulsing technique. 

 The results of pulse measurements on transducers 

 may be found in several reports. 55 



The procedures and problems involved in using 

 the pulsing technique vary with different types of 

 transducers. Careful thought is required before the 

 method is used in any particular case, and prelimi- 

 nary measurements are often helpful in determining 

 its applicability. The preceding discussion should 

 serve as a guide rather than as a rule in making pulse 

 measurements. 



