118 



mulas developed for steady flow, its numerical value at the given 

 velocity in the section, and giving it the sign of the velocity. 



CURRENTS PRODUCED BY A SIMPLE HARMONIC FLUCTUATION OF THE 

 SURFACE HEAD AND SLOPE IN A SHORT SECTION OF A CHANNEL 



237. The surface head in a section of a tidal channel is the differ- 

 ence between the elevation of the tides at the ends of the section. 

 These tides are, as has been seen, the resultant of a number of har- 

 monic components, of various amplitudes, speeds, and phases, occa- 

 sionally modified by meteorological disturbances. Quite obviously, 

 the surface head likewise is the resultant of harmonic components of 

 the same speeds; but it does not follow that the amplitudes of the 

 components of the head are proportional to the amplitudes of the 

 tides. On the contrary, the amplitudes of the diurnal components of 

 the head may be, and frequently are, proportionally less than the 

 amplitudes of the diurnal components of tides of the mixed type ; and 

 overtides of the head often are relatively more important than those 

 of the tides themselves. Generally, the head in a short section of a 

 tidal channel during a single tidal cycle does not depart widely from 

 a simple harmonic fluctuation with the speed, m2, of the principal 

 lunar component of the tides. The currents derived from such a 

 simple harmonic fluctuation of the head often afford a sufficient indi- 

 cation of the strength and timing of the actual currents; and in any 

 case provide a basis for a determination of the currents resulting from 

 a head which has a given variation from the simple harmonic fluctua- 

 tion assumed. 



238. Relation oj surface head to a simple harmonic fluctuation of the 

 tides at the ends of a short section of channel. — If the tides at the ends 

 of the section are taken to have simple harmonic fluctuations of the 

 same speed, the head likewise has a simple harmonic fluctuation of 

 the same speed. Let and A be two stations on a tidal channel, at 

 such a limited distance, I, apart that at any instant the variation in 

 the velocity and slope between the stations is immaterial. Let: 



yo=Ao cos {at+ao) (133)- 



be the elevation of the water surface at 0, taken as the initial sta- 

 tion, and let: 



yi=Ai cos (at-\-ai) (134)' 



be the elevation of the water surface at A. 



