EXPLICIT AND TACIT ASSUMPTIONS 



267 



that the discussion to this point has glossed over the 

 fact that neither the outgoing ping nor the scattered 

 sound which reaches the transducer is really single- 

 frequency sound. No sound of finite duration can be a 

 pure single-frequency sound since the latter theo- 

 retically lasts an infinite time; it can be shown that 

 the relation between the time duration 8t of a ping 

 and the width 5/ of the frequency band making up 

 the pulse is approximately 



8fdt = 1. (63) 



In general, the relation between any time-dependent 

 signal F(t) and the frequency spectrum of the signal 

 is given by Fourier's integral theorem.' That is, the 

 signal can be written in the form 



F{t) 



\/2irJ- 



A(co)e'"'rfco, 



■\/2t- 

 where A (to) is determined by the equation 



^(") = ;7^/->®^ 



"dt. 



(64) 



(65) 



Equations (64) and (65) are just generalizations of 

 corresppnding equations applicable to Fourier series 

 of periodic functions. In these equations F{t) and 

 A (oj) are generally complex, and o can be interpreted 

 as equal to 27r/ where /is the frequency of the spectral 

 component corresponding to to. 



It is possible, therefore, to make a frequency 

 analysis of any given ping, using equation (65). This 

 frequency analysis can then be used to obtain a formal 

 solution of equation (1). For, because of the linearity 

 of equation (1), if the scattered sound reaching the 

 receiver as a result of emission of the continuous 

 sound e'"' is B{o>)e^\ then the pressure of the scat- 

 tered sound reaching the transducer as a result of 

 any given ping is 



v(t) 



\/2- 





A{co)B{ccr'edco, 



(66) 



where A{oi) is given by equation (65) in terms of the 

 pressure variation F{t) of the outgoing ping. It is 

 necessary to qualify equation (66). Because the 

 boundary conditions and the velocity of sound at any 

 point are changing with time, the scattered sound 

 which reaches the transducer is not a pure sound, 

 even though a pure sound e""' is emitted. Thus al- 

 though the pressure of the scattered wave can always 

 be presented as a Fourier integral of the form (64), 

 equation (66) is not rigorously true, if B{co) and A{o3) 

 are defined as above. However, for the purpose of 

 investigating the validity of the assumption under 



consideration, these effects of time variation can be 

 neglected, and equation (66) accepted as valid. In 

 addition, for simplicity, the velocity of sound c in 

 equation (1) can be assumed constant. 



By using equation (66), the dependence on time of 

 the pressure p{t) of the scattered sound reaching the 

 transducer can be calculated for pings of any length 

 and for various types of scatterers. If p{t) is plotted 

 as a function of the time following the emission of the 

 ping, it turns out that p(t) is always zero imtil the 

 sound has had time to travel to and from the nearest 

 point on the scatterer. In other words, scattering 

 from an individual scatterer actually does begin at 

 the instant that sound energy begins to arrive at the 

 scatterer. It is not possible to show in general that 

 scattering ceases at the instant the sound energy 

 ceases to arrive at the scatterer. However, for the 

 special case of an infinitely rigid spherical scatterer, 

 it is possible to show that the duration of the scat- 

 tered sound received from the sphere is the same as 

 the duration t of the outgoing ping, as long as the 

 relation 



D 



c 



(67) 



is satisfied, where D is the diameter of the sphere. The 

 significance of equation (67) is simple; it means that 

 the scattered sound will have the same duration as 

 the initial ping, provided that the travel time of the 

 sound across the sphere is negUgibly short compared 

 to the duration of the initial ping. This result is not 

 unexpected. 



The reason why no general proof can be given for 

 the validity of the second part of the assumption 

 under discussion, namely, that scattering ceases at 

 the instant sound energy ceases to arrive at the 

 scatterer, is easily understood. Any real not infinitely 

 rigid scatterer, such as a bubble, will have definite 

 resonant frequencies which will be excited by the 

 incident soimd, and the scatterer may continue to 

 radiate sound at its resonant frequencies long after 

 the ping has passed by. Also some sound may enter 

 the scatterer and be reflected back and forth inside 

 the scatterer a number of times before it is scattered 

 back toward the transducer. If the scatterer is large 

 and many such reflections are possible, the duration 

 of the scattered sound will be longer than t. Despite 

 these difliculties it can be argued that the assumption 

 can be regarded as vaUd. There is good reason to 

 doubt that resonant bubbles or other resonant scat- 

 terers play a large part in reverberation; in any case. 



