262 Mr. J. McCowan on the 



the lunar and solar semidiurnal tides ; and if we do not seek 

 to trace its variation from spring tide to neap, it will be 

 sufficient to replace these, as their periods are roughly equal, 

 by one, and take 



77 = 970 sin n£ (32) 



as representing the oceanic tide at the river-mouth ; tj deno- 

 ting the elevation of the surface above mean level, and 

 n = 27r)T the "speed" of the tide, which has the period T. 

 But if we want to consider the phenomena of spring and 

 neap tides we must take 



rj — 7] l siring -\-r] 2 sin 7i 2 t ; .... (33) 



where n l and n 2 are the speeds, and 2rji, 2r\ 2 the ranges of the 

 lunar and solar semidiurnals respectively. 



Consider the case of a river or long narrow estuary which 

 approximates in form to a straight channel of uniform rect- 

 angular section. Let the origin be taken at the mouth of the 

 river, and let x be measured from the mouth up. Then if the 

 oceanic tide at the river-mouth be given by (32), we see by 

 (10') that, writing rj for z — h, the elevation above mean level, 

 the tide-wave propagated up the river in consequence will be 

 given bv 



v = Vo smn{t-%/(3\/J(hT^)-k}, . . (34) 



w T here k is a constant to be determined from the mean rate of 

 efflux from the river. Thus if the natural velocity of the 

 stream towards its mouth be q, then by (ll')? noting that the 

 integration need only extend over the period, 



I r t 



k = q+ -ml 2 ^g(h-tr) sin 27rt/T)dt; 

 1 Jo 



Jc = q+Z-^\ \/h + r) sind .dd: 

 n Jo 



-♦^^fV 1 - 



k=Q + E^^-j. . . . (35) 



The limiting values between which k must always lie are 

 therefore given by 



Jc = q + 2^gh when Vo = 0, . (36) 



4,\/2 



k = q -\ vgk == q + 1 '8 V gk when r) Q = h ; . (37) 



7T 



