304 The Harmonic Analysis of Tidal Observations 



the vicinity of the shores and in shallow water, the amplitude of the tides 

 cannot be regarded any longer as small in comparison with the water depth. 

 Frictional influences will cause deformations of the purely harmonic form 

 of the tide curve, in the form of overtides. 



Borgen (1894) (see Hessen, 1928, p. 1) extended Darwin's method, where- 

 by a great deal of computation work is eliminated and which needs less ob- 

 servations. It is based upon a special way of grouping the material which per- 

 mits to emphasize each time one of the partial tides, whereas the others remain 

 completely repressed. The reduction of the tide curve is first made in ordinary 

 mean solar time, so that for each day we have a horizontal series of 24 values. 

 If we add up a series of n days, the S x and 5" 2 -tide which conform to solar 

 time, will be added, whereas the other components will cancel each other 

 to a certain extent. A certain period of n days can now be found, in which 

 the influence of a given tide, for instance of M 2 attains a positive maximum ; 

 then another period of n days in which the influence of this tide M 2 has 

 a negative maximum, whereas S x and S 2 are present in both series in equal 

 intensity. If we form the difference of the sums of the two series, S x and S 2 

 will vanish and the effect of M 2 appears to be doubled. This procedure is 

 repeated m times and the result of the m series summarized. If m is selected 

 correctly, the influence of all other components, with the exception of the 

 one which is wanted, can be repressed to the greatest possible extent. Ac- 

 cording to this method, we need 92 selected days to determine M 2 accurately, 

 whereas Darwin's method requires observations over a full year. 



This Borgen method is most elegant, but nevertheless it proves very 

 difficult in practice, because the calculation, etc. cannot be left to untrained 

 help. This is, in effect, the reason why it is seldom used. 



Another reduction factor should be mentioned which should be incorporated into the methods 

 of Darwin and Borgen in order that the results of the harmonic analysis of different intervals may 

 be compared with each other. Contrary to the more accurate evaluations of the tide potential by 

 Doodson, the influence of the regression of the nodes of the moon's orbit is not incorporated in 

 the earlier classical methods as an independent component tide, but rather as a slow variation 

 of the amplitude and phase of the other components. The amplitudes Pi,P», ...P n ... in (X. 8) derived 

 from observations contain functions: (a) of the angle J between the moon's orbit and the equator, 

 (b) functions of the obliquity (f ) of the ecliptic; and (c) functions of the inclination /' of the moon's 

 orbit to the ecliptic. Darwin proposed to reduce P to the average value of these functions. He takes 

 a certain point of reference of the moon's orbit and then applies to the amplitude a factor/. Then 

 P = fH and H is the average amplitude of the tide. H and x are the harmonic constants of the 

 considered tidal constituent. Tables of these factors for l//can be found for different values of J in 

 Darwin and Borgen, respectively. 



The harmonic constants of the most important components can also be 

 derived with a fair accuracy from shorter periods of observation. For this 

 purpose, one requires at least 15 or 29 days of observation. Doodson (1938) 

 has given instructions for working up the observations for such short periods 



