SENSITIVITIES 



13 



ity, equation ((i) applies and F = — z,.i'„. Solving for 

 t'„ from equations (()) and (1), and substituting the re- 

 sult into equation (H). the transmitting sensitivity is 

 obtained as 



.S'(R) 



m) 



i»>p 



4tt 



k 



z + 



-g(R)- 



(16) 



Thus the transmitting sensitivity depends on the fre- 

 quency / = u>/2tt, the density of the medium p, the 

 open-circuit mechanical impedance z , the electro- 

 acoustic coupling constant /.', and the radiation impe- 

 dance :,., as well as the integral of Green's function 

 over the surface of the transducer. 



Consider next the receiving sensitivity M. Suppose 

 that in the absence of the transducer from the me- 

 dium there is present a sound held ^ inc (R), whose 

 value at the position of the acoustic center of the 

 transducer R (when the latter is not present in the 

 medium) is p im .(R„). r Then the ratio of the open- 

 circuit voltage generated by the transducer when in 

 the medium to /> ilu .(R ) is defined as the receiving 

 sensitivity M. e 



The value of M depends upon the type of wave, 

 that is, p ini .. For practical applications, the important 

 case is that for which p hli . is a spherical wave with cen- 

 ter at some point R,.. The plane wave sensitivity can 

 be considered to be the sensitivity to spherical waves 

 when R,. is infinitely distant from R (l . In this case p Uu . 

 can be written as 



[R R] 



frnc(R) = * ' 



R - R,| 



(17) 



The actual pressure present at a point R when the 

 transducer is in the medium must now be found. If 

 the transducer diaphragm did not move, the pressure, 

 from the definition of Green's function, would be 



*G(R, R,.). 



On the other hand, if no incident sound pressure p im . 



t The acoustic center is the center of symmetry if it exists, but 



for this discussion it may be any arbitrarily cljosen point on the 



transducer. The incident sound held pressure />. satisfies the 



1 l inc 



wave equation y + ( — J p (R) = 0, where X 



wave length and y2 j s the Laplacian operator. 



e In practice the receiving response is always given for a uni 

 form plane wave normally incident on the transducer. 



the 



is present in the medium but the transducer surface 

 has a velocity v n , the pressure at a point R in the 

 medium is as given by equation (14). If these two pres- 

 sures are added, the actual pressure at R, p(R), when 

 p iDC is present, is obtained and the transducer dia- 

 phragm has a velocity v n . Thus 



p(K) = <t>G(R, R,J 



]-'P 



]"'(> 



G(R,r)dr (18) 



= *G(R,R,.)-^„g(R). 



Since the total force on the transducer diaphragm is 

 the integral over the diaphragm of the pressure at 

 each point on its surface, then 



F = 



p(r) dr = * / G(r, R,.) c/r 



1">P 



g(r) dr. 



;i9) 



Since G(r, R,.) = G(R,„ r) from the symmetry of 

 Green's function, the first term can be written as 

 <i>g(R,.). If we compare equation (19) with equation 

 (5), we see that 



F n 



*£(*.■) 



(20) 



and 



= g| g g(r)dr=g/jf G (r,rOdr'dr. (21) 



These quantities may be computed when Green's 

 function for the surface S is known.'' 



If (5) is substituted for F, with /*' and z r given by 

 equations (20) and (21), in the fundamental equa- 

 tions (1) and (2) and the case where the transducer is 



lilt can further be shown that, for an arbitrary incident 

 sound held /). , one has 



1 r 2)b. (r) 



<>(R) = />,„ (R) - — / ' ■"' - G(R,f 



/'<R) = /> jn ,.(R) 



-J, 



dn 



i(R.r) dr 



■7^ f G(R, 



r) dr. 



The first two terms reduce to cj) GfR. R ,) as shown in equation 

 (18), if p. is the spherical wave of equation (17). If p(R) is in- 

 tegrated over the surface S ,, the integral of the first term />. is 

 what /■'. was called, while the integral of the second term is 



what /•" 



was called. The last term is again — z. 



