300 The Harmonic Analysis of Tidal Observations 



computed from the orbital elements of the tide-generating bodies, which 

 vary in relation to the mean time, x/a is the lag between the tide and the tide 

 potential, respectively, the time of the high water interval, x is called the phase 

 of the partial tide in the locality in question or kappa number. H is the mean 

 value of the amplitude or semi-range of the component or partial tide. The 

 epoch x is the difference between the theoretical and actual phase as determined 

 from the tidal observation. H and x are functions of localities on the earth 

 and are different for each harbour. It is these constants which must be derived 

 from the tidal observations. They are called the harmonic tidal constants 

 of a locality. If they are known, together with the diurnal part of the partial 

 tide V , the above-mentioned formula permits computation of, for any time t 

 of the day, the deviation r\ of the surface from mean level and, if the height 

 of mean level above chart datum is known, the water depth at that time. 



The main purpose of the harmonic analysis consists in deriving from the 

 heights of the observed tides in a locality over a great length of time (sea- 

 level recordings or hourly observations) the harmonic constants H and x for 

 each component of the tides. The observations should be considered as the 

 superposition of a large number of waves of the form (X. 1), each with differ- 

 ent H , a and x. Each one of these waves repeats itself in the same way in 

 the time interval T = 2nja. Let us consider such a wave with the period To 

 and let us divide the tidal curve into equal parts of this length T . For instance 

 in the case of the M 2 tide, in two periods of 12 and 24 lunar h respectively 

 (counted from the transit of the moon through the meridian in the locality) 

 high and low water will always occur in each of these equal parts at the same 

 hour. We read from the tidal curve the height of the sea level for each lunar 

 hour, and if we take for each particular hour the average height over a very 

 long series of lunar days, the average variation in the sea level of such a lunar 

 day will be particularly apparent. At the same time, all other components 

 which have different periods check and counteract each other; therefore, if the 

 series of observations extends over a large number of lunar days, these different 

 components balance each other out. Thus, we can eliminate all other waves, 

 with the sole exception of the one under consideration, and the mean varia- 

 tion in sea level obtained in this manner can be regarded as that of the consi- 

 dered partial tide for one single day of observation. One-half of the range is the 

 amplitude, whereas the interval between the transit of the fictitious "moon" 

 through the meridian and the high water gives the phase of the partial tide. 



The observations show the sea level rj as a function of time which can be 

 written in the form 



r 



tj = H cos(a t—x )+ 2j H n cos (o n t—x n ) = 



r 



= B cos a t + C sin at + \ H n cos {o n t — x n ). (X. 2) 



