12 ZEILON, ON T1DAL BOUNDARY-WAVES. 



ignored for the moment. As usual in this kind of applications of Cauchy's theorem the 

 integration contour will be completed by an infinite semicircle round the northern or 

 southern half of the complex Å>plane; this semicircle being supposed to be the limit of a 

 series of semicircles avoiding always to pass through the imaginary roots of D. 

 For an x positive and great enough to make 



l— x<0 



for all values of X for which 



the integration is to be conducted round the southern part of the fc-plane, and will give, 

 since the direction of integration is negative: 



+ _ icoQKjgK — a 2 coth Kh') 



sinh Kh 



— o* coth A k ) .. . ,, . /» ... ■„. ,. 

 '\di)k-+K J 





— 03 



= A sin (ut — K x) + a sin (o^ — •/ x) 



As to r lh , it is immediately found to be: 



II: 2 >7+= J 40(ir)sin (ot — Kx) + a(-)U) sin (a t — v.x) 



For a negative x again the integration must be conducted round the northern half- 

 plane, and the result will be: 



II: 3 jjjj" = — A sin (a t + Kx) — a sin (a t + y.x) 



11:4 r lh> ==z ~ AQ(K) sin (at + Kx) — aQ(x) sin {o t 4 *x) 



These formulce represent trains of sim/ple-harmonic waves, progressing rightwards for 

 the positive half of the axis of x, and leftwards for the negative half of it. 



For the roots K and /. entering here, denoting by K the smaller root corresponding 

 to the greater wave-length, it is easily found that K will be very small and approximately 

 given by: 



K^m-Ml + D^ '- 



or 



o 



- Vg(h + h') , 



whereas the other root x, on account of the smallness of o, will, with sufficient exactitude, 

 be the real root of the equation: 





