18 ZEILON, ON T1DAL BOUNDARY-WEVES. 



As for the real roots, writing: 



^ ( " /) ^(7/l),_--^(sinkl + sln^)-^-^ 

 we obtain the residue: 



2% C/.GQ 



__ : i g«(o/ — y..r) I f -ri 2 /.-g»z/ 



\/ n sinh xh .^/,(x) ' J 



dl 



or denoting by a the positive constant 



+ 00 



2 i CKffp ., , , . /" , ., . , _. 



— — — -i-y-.fi"^") e-"-' ? t~ l -'di. 

 y.h. _/,(/.) / 



C/.GQ 



and taking the real parts: 



sinh y.h . J x {v.) ' 



X 



2 a ■ ' ■ C 



— sin (g i — / .r) e~ " 2 ** cos a k dl 



Vit J 



— oc 



QC 



2a / ' 



+ —= sin (Gt + 7.x) e~" 2 )? cos /Idl 



V 71 I 



2a C 

 -— cos (Gt — -ax) I t~ " 2; - 2 sin /.Idl 



V7t . 



2a ' 



- Ti cos (a* + **) \ e~" 2; - 2 sin x/.rf/ 



From the relation: 



4-a; 



I e-" 2/ - 2 sinxArfA = 



it follows tliat the coefficients of the two cosine-waves are numerically equal and of opposite 

 signs, so these waves will combine into a standing one: 



2a . . C ' .-i". ' ,'_,, 

 . . sin at . sin •/.% . I e _ " '• sin //.rf/. 



1 •' J 



As a whole, then, the disturbance dne to the ridge will be made np from the follow- 

 ing three constituents: 



I. a wave diverging from x = o to the right, with an amplitude increasing with x 

 from the value 



