IV. HARMONIC ANALYSIS (cont.)

27. Harmonic Analysis for Tides of Long Period

For the purpose of determining the tides of long period we have to eliminate the oscillations of water-level arising from the tides of short period. As the quickest of these tides has a period of many days, the height of mean water at one instant for each day gives sufficient data. Thus there will in a year’s observations be 365 heights to be submitted to harmonic analysis. To find the daily mean for any day we take the arithmetic mean of 24 consecutive hourly values, beginning with the height at noon. This height will then apply to the middle instant of the period from 0^h to 23^h – that is to say, to 11^h 30^m at night. The formation of a daily mean does not obliterate the tidal oscillations of short period, because none of the tides, except those of the principal solar series, have commensurable periods in mean solar time. A small correction or "clearance of the daily mean," has therefore to be applied for all the important tides of short period, except for the solar tides. Passing by this clearance, we next take the 365 daily means, and find their mean value. This gives the mean height of water for the year. We next subtract the mean height from each of the 365 values, and find 365 quantities δh, giving the daily height of water above the mean height. These quantities are to be the subject of the harmonic analysis, and the tides chosen for evaluation are those which have been denoted above as Mm, Mf, MSf, Sa, Ssa.

where t is time measured from the first 11^h 30^m. If we multiply the 365 δh’s by 365 values of and effect the summation, the coefficients of B,C,D, &c., are very small, and that of A is nearly . Similarly, multiplying by we obtain 10 equations for A, B, C, &c., in each of which one coefficient is nearly and the rest small. These equations are easily solved by successive approximation. In this way A,B,C, &c., are found, and afterwards the clearance to which we have alluded is applied. Finally the cleared A,B,C, &c., are treated exactly as were the components of the tides of short period. Special forms and tables have been prepared for facilitating these operations.







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