U. S, COAST AND GEODETIC SUE^^EY. 



2. GENERAL EXPLANATION. 



A simple harmonic function is a quantity that varies as the cosine 

 of an angle that increases uniformily with time. In the equation. 

 y = A cos at, y is an harmonic function of the angle at, in which a 

 is a constant and t represents time as measured from any initial 

 epoch. 



Harmonic analysis as applied to the tides is a process by which 

 the actual observed tide at any place is separated into a number of 

 partial or constituent tides of which it is composed, the rise and fall 

 of each partial tide being a simple harmonic function of time. 



Harmonic prediction of the tides consists in reuniting the partial 

 tides in accordance with the relations which will prevail at the time 

 for which the predictions are to be made. 



The partial tides are called components and are usually repre- 

 sented by letters either with or without subscripts, as M2, Kj, Mm, and 

 Sa. Theoretically, the tides consist of innumerable components of 

 various magnitudes, but only a comparatively few are of sufficient 

 size to be of practical importance in the prediction of the tides. The 

 predicting machine used by the Coast and Geodetic Survey is de- 

 signed to take account of a maximum of 37 components. 



Each component represents an elementary periodic cause produc- 

 ing or affecting the tide. The principal component, designated as 

 M2, represents the mean effect of the moon. Another component 

 S2, represents the mean effect of the sun. Other components take 

 account of the various inequalities in the motions of the moon and 

 the sun, such as changes in parallax and declination, and also inequali- 

 ties resulting from shallow water and seasonal meteorological changes. 



The amplitude of a component, commonly designated by the 

 letter H, is the semirange between the maximum and minimum 

 heights of the tide due to that component. The amplitude of any 

 component varies with the locality, but for any particular place it is 

 practicalh^ constant for all time. 



The epoch of a component, commonly designated by the Greek 

 letter kappa (/c), is an angle whose value depends upon the interval, 

 between the time of the maximum of the component as determined 

 theoretically from the equilibrium theory and the actual time as 

 determined from the analysis of the observations. The epoch of a 

 component varies with the locality but, like the amplitude, is con- 

 stant for any particular place. 



The harmonic constants are the numerical values of the amplitudes 

 and epochs of the components for any place. The determination of 

 these constants from the records of tidal observations is the purpose 

 of the harmonic analysis. 



The rise and fall of the tide may be graphically represented by a 

 curve, with the ordinates representing the height of the tide and the 

 abscissas the time. The tidal record as traced by an automatic tide 

 gauge is such a curve. The general equation of this curve, giving 

 the height of the tide as a function of time, is usually written in the 

 form 



y = Ho + A cos {at + a)+B cos (ht + /3) + C cos (ct + 7) + etc. (1) 



in which y is the height of the tide at any time t. Ho is a constant 

 depending upon the datum from which the heights are reckoned, and. 

 each cosine term represents the height of a component tide. 



