﻿Resonators exposed to Primary Plane Waves. 221 



persistence that we may calculate the pressure as if they 

 were permanent. Thus if yfr be the velocity-potential, we 

 have as before with sufficient approximation 



, . 1 — ikr 1 d-dr 1 



M a = •> - i = — * ; 



r a ar r~ 



so that, if p be the radial displacement of the spherical 

 surface, dp/dt = —a/r 2 , and 



yfr=-r(l-ikr)dp/dt (47) 



Again, if <r be the density of the fluid and Sp the variable 

 part of the pressure, 



Sp = - adyjr/dt = vr{l-ikr)d 2 p/dt 2 , . . (48) 



which gives the pressure in terms of the displacement p at 

 the surface of a sphere of small radius r. Under the circum- 

 stances contemplated we may use (48) although the vibra- 

 tion slowly dies down according to the law of e mt , where n 

 is not wholly real. 



If M denotes the " mass " and fx the coefficient of resti- 

 tution applicable to p, the equation of motion is 



Mj+ W + .iW(l-ttr)g=0, . . (49) 



or if we introduce e int and write M 7 for M + 47ro-r 3 , 



n s( _ W + iirtrkr* . i) + fi = 0. . . (50) 

 Approximately, 



n = y/iixjW) .{!+*. %ir<rfo*p&! } ; 

 and if we write u=p + iq, 



p=y/(p/W), q^p.^TrakryW. . . (51) 



If T be the time in which vibrations die down in the ratio 

 of «:1, T=l/j. 



If there be a second precisely similar vibrator at a distance 

 li from the first, we have for tho potential 



+--R'- M t. (52) 



and for the pressure due to it at the surface of the first 

 vibrator 



£ a) ' 2 il-Tt d 2 02 ¥ , 



