324 



VT f )9 , and t? should bear the superscript (1) indicating that they 

 are coefficients of ot in the oc -expansions. We shall omit the super- 

 scripts here. 



It should be mentioned at this point that the theory of this 

 section is worked out on the Hs?umption that cavitation does not occur. 

 Mathematically speaking, this means that Eqs. (1) and (2) are assumed to 

 hold everywhere in the fl\iid. Cavitation is discussed briefly and quali- 

 tatively in Section 2 of this Part. To avoid cavitation it will be shown 

 there that it is necessary (but not sufficient) to confine ourselves to the 

 case of circularly clamped shells. 



Getting back to the main current of the argument, it is known-' 

 the solution of (2) at a point in a volume (at every point of vrtiich (2) 

 is satisfied) can be expressed as the sum of the retarded potentials due 

 to distributions of sLnple and double sources on the surface of the volimie. 

 Consider two semi-infinite volumes above and below the r-ji^-plane (in which 

 the plate and other diffracting objects, if any, lie). Moreover, consider 

 the velocity potential "^ in each semi-infinite volume to vanish at infinity, 

 and consider the "VJA in one voliome to be given by the reflection through the 

 »^<fc- plane of the "U/ in the other volume. Under these conditions it fol- 

 lows that the 'Vj'' at a point "? below or on the r,^-plane can be e:jq3ressed 

 in the form 



t* = t - \r*-f"\/Co ^ (4) 



T/ Kirchhoff "Zur Theorie der LicHtstrahlen, " Berl. Ber. 13^2, p.. 641 

 /pes. Abh. ii 22j. 



15 



