28 ZEILON, ON TIDAL BOUNDARY-WAVES. 



for the first terms, and, at any råte, for great values of v. 



V.y > t'/.. 



For a positive and rather great value of x the residue theorem in the usual way gives 

 (only terms contributing to the proper boundary-wave being written): 



; _ 2 . y r ^ Q (g, v - r^coth ^) ^...^ M fj {l) ^i dL 

 v-i smhxji.l--) \* IJ 



Here we introduce: 



/(*) = , e-' 



Pvb.)? 



1 . 



/ 



This represents a ridge which, to give a sensible disturbance, is extended över a definite 

 portion i 

 for mula: 



portion of the wave-length — of the boundary-wave of period — . According to a known 



' 4 a 2 y. z 



VttJ <** 



Because^of the relation 



-/,, .> i'/., 



it follows that the series obtained 



U % - _ 2 ^ V * ° S * ( ** ~ V * *$\** K) sin (vot-w), J v Ml 



will converge rapidly, so that only terms corresponding to rather small values v will be 

 of any importance. The same would also evidently hold true for terms corresponding 

 to the roots K v . The convergence will be further emphazised by the following conclusions. 

 Physically speaking, a small c is one that is small compared to the boundary-wave- 

 velocity. That is: 



= zc 

 o 



is then a small number. 



Assuming the parameter a to be small enough to make all terms insensible but for 

 a few ones in the beginning, for which 



•/„ = VY., 



then the coefficients of the different sine-functions will be found to be of the order of 

 magnitude: 



