102 Lord Rayleigh : Further Applications of Bessel's 



grazing, in which case the usual formulae * become nugatory. 

 It will be convenient to fix ideas upon the case of sonorous 

 waves, but the results are of wider application. The general 

 differential equation is of the form 



dt*~ y \M* dy*)> W 



which we will suppose to be adapted to the region where as 

 is negative. On the right (x positive) V is to be replaced 

 by Vx, where V 2 >V, and (f> by fa. In optical notation 

 Y 1 /Y=fi, where //, (greater than unity) is the refractive index. 

 We suppose <j) and <f> x to be proportional to tftto+c*), b and c 

 being the same in both media. Further, on the left we 

 suppose b and c to be related as they would be for simple 

 plane waves propagated parallel to y. Thus (4) becomes, 

 with omission of e^ bi/+ct \ 



g=0, **f-*0,-l>/* . . . (5) 



of which the solutions are 



<£=A + B.r, ^ 1= Cr ia! ^ 2 - 1 ^ ... (6) 



A, B, G denoting constants so far arbitrary. The boundary 

 conditions require that when «t = 0, d^/dx^dfa/da: and 

 that pcp = p 1 (j) 1 , p, pi being the densities. Hence discarding 

 the imaginary part, and taking A = l, we get finally 



^l 1 - ^" 1 ' } 008 '^ 4 • • (7) 



<!>!= V-e- 6 *^ 2 - 1 ^ cos (by + ct). .... (8) 



Pi 



It appears that while nothing can escape on the positive 

 side, the amplitude on the negative side increases rapidly as 

 we pass away from the surface of transition. 



If yu,<l,awave of the ordinary kind is propagated into 

 the second medium, and energy is conveyed away. 



In proceeding to consider the effect of curvature it will 

 be convenient to begin with Stokes's problem, taking advan- 

 tage of formulae relating to Bessel's and allied functions of 

 high order developed by Lorenz, Nicholson, and Mac- 

 donaldf. The motion is supposed to take place in two 

 dimensions, and ideas may be fixed upon the case of aerial 



* See for example ' Theory of Sound/ vol. ii. § 270. 

 t Compare also Debye, Math. Ann. vol. lxvii. (1909). 



