512 Sir G. Greenhill on the 



where U 2 =#/t, and U is the velocity and TU the wave length 

 of the long flat tidal wave in open water of depth h 



Taking an average depth of water of 12 fathom makes 

 U over 48 f/s, nearly 30 knots, and the semi-diurnal wave 

 length 12 X 30 = 360 miles, geographical (G). 



Then ^ = 0, and there is no rise and fall of the tide, 

 where xly/(lh) = 2*4, the first root of J = 0. making 

 „'<• = 137 G miles from the end of the estuary. 



And f = 0, and there is no tidal current where x/y/(lh) = 3*8, 

 the first root of Ji = 0, and # = 218 G miles. 



These figures may be taken to apply to the tide in the 

 Bay of Fundy. 



In an estuary, shallowing to nothing uniformly at one in rt, 

 and given by b = ca? m in plan, A = cx m+1 /n, 



d / ,,dm\ )i.r" l ri d?n . -^dn nri ~ 



■dx\ dxj I dx\ v J dx I 



fl = C.(y), • (10) 



l l 2 £ / n> d£ nP ,' . „ /n#\ ^ n , 



^l + (" l+2 - , i + T = °' f = c- +1 ( T ); <ii) 



and the previous considerations may be resumed. 



Thus with m = ^, and parabolic in plan, the solution 

 is given in finite terms by Ca and Cs. 



For a canal of uniform breadth, ??i = 0, and shallowing 

 uniformly, <?=1 ; aud then 



With m = 0, ? = 2, the D.E.'s for f and 77 have 



an algebraical solution; and with A = 67i(l 2 1, rj is 



given by PJ - j. 



Generally, with m = 0, and (-) =(2 — q) 2 z, 



and with uniform depth h, m = q, and I - ) = (q— 2fz. 



•S^( i+ J^s)S + — °« "= c »^ -H^- (13) 



In all these cases the vertical cross section of the channel 

 may be taken rectangular or elliptical, but it ought to be 

 supposed to vary slowly. 



