HABMONIC ANALYSIS AND PREDICTION OF TIDES. 3 



A single component tide referred to its mean level as the datum is 

 expressed by the equation 



yi=A cos (at + a) (2) 



in which y^, the height of the component tide, is a function of t, the 



time reckoned from some initial epoch. 



The coefficient A is the amplitude or semirange of the component. 



The angle at + a changes at the rate of a units of angle per unit 



of time, and this rate of change is called the speed of the component. 



The period of the component is the time required for the angle 



360° 

 at-\-a to go through a cycle of 360° and is therefore equal to 



when a is expressed in degrees. The phase of the component at 

 any time t is the value of the angle at + a, with multiples of 360° 

 rejected, or it may be defined as the angular change in the component 

 since the time of the preceding maximum or high water of the com- 

 ponent. The initial phase is the phase at the instant from which 

 the time is reckoned; that is, when ^ = 0, and is equal to a in the 

 above angle. 



A component tide is also expressed in the following forms : 



yi = R cos {at - ^ (3) 



y, = fHco^[{V-Vu)-K\ (4) 



T/i = fH cos [at + ( Fo + u) - k] (5) 



In an analysis theoretically perfect the coefficient A of formula (2) 

 must be an absolute constant; but in practice it has been found 

 convenient, in order to take account of the effects due to the changes 

 in the longitude of the moon's node, to consider this coefficient as 

 subject to certain variations. These variations are, however, so 

 slow that for a series of observations not exceeding a year in length 

 the coefficient may be treated as a constant, but factors are applied 

 for reducing the results from different years to a mean value. 



The coefficient R of formula (3) represents the unmodified ampli- 

 tude applying to a particular series. The mean value of the ampli- 

 tude for all years is represented by the H of formulas (4) and (5). 

 The /is a factor, usually near unity, which gives the theoretical rela- 

 tion between the observed amplitude from any series of observations 

 and the mean amplitude, this relation depending upon the longitude 

 of the moon's node. 



The angle C, of formula (3) is the equivalent of —a of formula (2). 

 The angle (V-\-u) of formula (4) is the theoretical phase of the com- 

 ponent for any time t as derived from the equilibrium theory, and 

 the epoch k is the difference between the theoretical and actual 

 phase as determined from the tidal observations. The complete 

 angle {V+u) — k is the equivalent of the angle at + a. in formula (2). 



The angle {Vo + u) of formula (5) is the value of the angle {V+u) 

 when t equals zero. The change in {V+u) being the speed of the 

 component, its value at any time t is equal to at + {Vo + u), and 

 formula (5) is therefore equivalent to formula (4). The values of 

 the mean amplitude H and the epoch k of the above formulas are 

 the harmonic constants which are to be determined by the analysis. 



