230 



Multiplying equation (306) by cos and equation (307) by sin and 

 adding: 



An sin (t>—aB/g=-0 (308) 



441. The equation of continuity is (equation 183): 



b{vzD)/c)x-{-z^y/^t=0 

 Wlien the depth, D, is constant, this equation becomes: 



Dvbz/bx-\-Dz()v/dx+zc)ylc)t=0 (309) 



Substituting the differential coefficients and the expression for v, 

 from equations (303) and (304): 



—DB sin {at—nx—<l>)dz/bx-\-nDBz cos (at—nx—<f) 



— a Az sin {at— nx)=0 (310) 



The equations of condition, derived by placing at—nx=0 and 

 at—nx=Tr/2, are: 



DB sin cl>c)z/bx+nDBz cos (l)=0 (311) 



—DB cos <i)C)zlbx+nDBz sin <l)—aAz=0 (312) 



Multiplying equation (311) by cos 4> and equation (312) by sin 4>, 

 adding and dividing by z: 



nDB—aA sin <t>=0 (313) 



Combining equations (308) and (313) to eliminate sin 0: 



aB/gAn ^ nDB/aA 

 whence: 



n^=aygD n = al^l^ (314) 



The rate of advance of the tide and current in an ideal estuary is 

 then -yjgD, the rate of advance of a frictionless progressive wave. 



442. From equation (311): 



bzlz=—n cot 4>bx 



The integration of which gives: 



2=i^e-"^^°*^ (315) 



in which K is the constant of integration. AMien x=0, K=z. K is 

 then the width of the estuary at the entrance, which conveniently may 

 be designated Zq. Then: 



z=Zoe-"'='°^'^ (316) 



Or: _ 



log 2=log Zo—{ax cot cj) log e)l-yJgD (317) 



If, then, the depth of an estuary is constant, and the width varies in 

 accordance with the law expressed by equations (316) or (317), its 



