106 Lord Ray] eigh : Further Ajijili cat ions of BesseVs 



is moderate, the escape of energy must be very small, and 

 accordingly that the vibrations inside have long persistence. 

 There is, however, something to be said npon the other side. 

 On account of the concentration near the reflecting wall, the 

 store of energy to be drawn upon is diminished. At all 

 events the problem is worthy of a more detailed examination. 

 Outside the surface of transition (r=a) we have the same 

 expression (9) as before for the velocity-potential, k and V 

 having values proper to the outer medium. Inside k and V 

 are different, but the product kY is the same. We will 

 denote the altered /: by h. In accordance with our suppo- 

 sitions h>k, and h/k represents the refractive index (//,) of 

 the inside medium relatively to that outside. On account of 

 the damping k and h are complex, though their ratio is real ; 

 but the imaginary part is relatively small. Thus, omitting 

 the factors e ilcYt cos nd, we have (r>a) 



4>=AB n (kr), (28) 



and inside (r<a) 



4>=BJ x (hr) (29> 



The boundary conditions to be satisfied when ?' = <x are 

 easily expressed. The equality of normal motions requires 

 that 



kAB n , (ka)=hBJ n '(ha); . . . . (30) 



and the equality of pressures requires that 



trAD n (ka)=pBJ n (ha), .... (31) 



<r, p being the densities of the outer and inner media respec- 

 tively. The equation for determining the values of ha, ka 

 (in addition to h/k = fi) is accordingly 



*JV(kp hJ n '(ha) 

 ~<TD n (ka) - pJjha)' 



Equation (32) cannot be satisfied exactly by real values of 

 h and k ; for, although J n 'jJ n is then real, iV/D n includes 

 an imaginary part. But since the imaginary part is rela- 

 tively small, we may conclude that approximately h and h 

 are real, and the first step is to determine these real values. 



Since ka is supposed to be decidedly less than n, D n and 

 Dn are given by (18), (20) ; and, if we neglect the imaginary 

 part, 



¥P = -sinh/3 (33), 



SJ n {ka) 



