Functions of Mgli order to the Whispering Gallery. 101 



A comparison with the simple case where the surface of 

 die vibrating body is plane (;?; = 0) is interesting, especially 

 iis showing how the partial escape of energy is connected 

 with the curvature of the surface. If V be the velocity of 

 propagation, and 2irjk the wave-length of plane waves of the 

 given period, the time-factor is e ikYt , and the equation for 

 the velocity-potential in two dimensions is 



3+0+**-° w 



If 4> be also proportional to cos m//, (1) reduces to 



+(*»-«■)*=©, (2) 



of which the solution changes its form when m passes 

 through the value l\ For our purpose m is to be supposed 

 greater than /,:, viz. the wave-length of plane waves is to be 

 greater than the linear period along y. That solution of (1) 

 on the positive side which does not become infinite with x 

 is proportional to e -xV (.■»#-&) y S o that we may take 



$= cos JcYt . cos my . ^VO" 2 -*-). . . (3) 



However the vibration may be generated at x = 0, provided 

 only that the linear period along y be that assigned, it is 

 limited to relatively small values of x and, since no energy 

 -can escape, no work is done on the whole at o? = 0. And 

 this is true by however little m may exceed k. 



The reason of the difference which ensues when the 

 vibrating surface is curved is now easily seen. Suppose, 

 for example, that in two dimensions <£ is proportional to 

 ■cos n#, where 6 is a vectorial angle. Near the surface of a 

 cylindrical vibrator the conditions may be such that (3) is 

 approximately applicable, and cf> rapidly diminishes as we go 

 outwards. But when we reach a radius vector r which is 

 sensibly different from the initial one, the conditions may 

 •change. In effect the linear dimension of the vibrating 

 compartment increases proportionally to r, and ultimately 

 the equation (2) changes its form and <f> oscillates, instead 

 of continuing an exponential decrease. Some energy always 

 escapes, but the amount must be very small if there is a 

 sufficient margin to begin with between m and /r. 



It may be well before proceeding further to follow a little 

 more closely what happens when there is a transition at a 

 plane surface x~0 from a more to a less refractive medium. 

 The problem is that of total reflexion when the incidence is 



