Theory of Long Waves. 263 



so that in approximations the value given by (36) may 

 generally be taken. 



By means of these results we can easily see the chief 

 characteristics of the tidal ebb and flow in such a river or 

 estuary. From (34) it follows that the height of rise and 

 fall is the same at all stations on the river as that of the 

 oceanic tide at the mouth, but the rise and fall is not simple 

 harmonic except at the mouth. We see also that high water 

 occurs later at stations up the river than at the mouth, the 

 lag, which is simply proportional to the distance from the 

 mouth, being less for high than for low tides, and also less 

 the less the natural flow in the river : the same is true for 

 the time of low water, but the lag is greater and it is greater 

 for high than for low tides ; hence the interval from low 

 water to the high water following is less than the interval 

 from high water to the following low water. In fact, from 

 (34), the interval from high water to the succeeding low 

 water, at a station at distance f from the mouth, is 



* T + H3^(^ )^~3v/ ;? (/i+ % )-J' (38) 

 or approximately, taking k=q-\-2\^gh and neglecting ?/ 2 , 



iT+fffef ; (39) 



and therefore, making the same approximations, the interval 

 from low water to the following high water at the same 

 station will be 



> l \^i- q ? (40) 



From (39) and (40) it follows that the difference in the 

 two intervals increases directly as the distance from the 

 mouth and as the height of the tide approximately : it is 

 also greater the greater the natural velocity q of the stream. 



Again, by (9'), 



u=2Vg(h + v )-k, (41) 



whence it follows that the rate of flow is a maximum at high 

 water, and the rate of ebb a maximum at low water. Also, 

 flood ends and ebb begins, or vice versa, when 



2s/g(h + V ) = k, (42) 



and therefore at an elevation above mean level which is 



