20 



ZEILON, ON TIDAL BOUNDARY-WAVES. 



The roots /,. themselves will be most easily computed graphically, and the corresponding 

 series of residues will be not too badly convergent. Writing 



sin (at + ax) = — sin (af — v.x) + 2 sin at cos xr, 



we construet the auxiliary curves: 



4a C 



'dl 



lOQy • '/c 



2 



! i ' 



i 











.? ^i^ 



.r ^^ 



i jr ~t 



~\l \ 



-å 5 



§ s. 



r S. 



i ^ 



f ^C 



■ Å ^ ^ 



/ S 



i 1 ^^L 



y S ^ 



r ^ y 



/ ^^ 



JL 3 



/ \* ^** 



i\ ^^* 



m "*^^ mt 



113 4 5 



Fig. 1. 



2/2 ~ A ccsinhh.jtdh) ■ r v.J ' i7,J -I' 



2 a 



and finally have: 



/; = — 2 - sin ((/ f — x #) + ?/, . sin t; £ cos kjc + y 2 . sin at. 



The curve 1. is in an obvious manner related to the error-integral; as for the second curve, 

 it is shown in the above diagram, with, of course, an exaggerated vertical scale, for 

 the case: 



a = i, a = 2, h =i h' = 4, q*— q' = O.oie, q = 1, g = 1,000, 



the units being centimeters and seconds. 



In this manner the five wave-profiles of fig. 2 below were constructed. In order 

 to show the wave-form as clearly as possible, the upper water-layer is painted black, and 

 beneath it the ridge shows the familiar silhouette of the probability curve. For gain of 

 space only half of the black water is shown, so that the free surface should strictly be 



