or after the epoch the two lunar periods will counteract each other 

 (viz. tend to raise to separate waves instead of one in common) 

 because a periodically acting agent which comprises 12 periods in 

 a lunar year can not continue to coincide with an agent that comprises 

 13 periods in the same space of time. A regular formation of waves 

 all the year round cannot be expected. 



3) It may, however, be expected to recur the same season next 

 lunar year, though with some discrepancy, seeing that the two lunar 

 periods are not quite synchronous after 355 days, but differ some 

 0.61 days. In reality the discrepancy is much greater because we 

 have omitted to count the influence of the third or anomalistic 

 lunar period. This period is 27,55 days and tends to cause a displace- 

 ment of 7 instead of 10 days from year to year in the wave-series. 

 This period is also more irregular than the others. 



It is only under favourable circumstances when two lunar pe- 

 riods combine that we may expect welldefined moon-waves at inter- 

 vals of 7 or 14 days to arise in a certain part of the year. These 

 wave-series may be expected to reappear at the same season next 

 lunar year and even in the following, though the discrepancy with 

 the solar year will have increased. After that, however, the analogy 

 will very likely cease as was the case in the year 1912. 



This brings us to a fact that hitherto has been overlooked, and 

 yet characterises every phenomenon of cosmic origin in which the 

 lunar periods and the tide -generating force play a part, viz. 



That every periodic phenomenon of this kind has a limited term 

 of existence. The period is »worn out » in consequence of the inner 

 want of harmony between the different lunar periods. Another conse- 

 quence is: 



That no periodic phenomenon which depends on the tide-generating 

 force will reappear in a perfectly analogous manner within finite time. The 

 period repeats itself approximately a given number of times till it 

 disappears and a new combination of periods prevails. 



These corollaries may surprise those who look upon cosmic 

 phenomena as possessing a continuous unchanging periodicity. It is 

 commonly supposed that the periodic nature of a phenomenon 

 is only proved by its regular repetition through endless time. In 

 studying its periodicity by harmonic analysis it is usual to found the 

 analysis on as long a series of observations as possible, so f. inst. 

 on observations of the frequency of sunspots, of the waterlevel, pre- 

 cipitation, the mean-temperature in different places etc. As a fact, 

 however, a phenomenon in Nature may be of perfectly periodic type 

 and yet have a limited term of existence. 



Every lunar period still exercises its influence, but no longer 

 in conjunction with the others. The big moon-waves break up 

 into minor waves which succeed one another with apparent 

 irregularity, not always at intervals of 14 days, but sometimes less. 

 Thus they become impossible to distinguish from the short tidal 

 waves and from local submarine seiches in the border-layer of the fjord. 

 If the level of this layer remained constant and the specific gravity 

 of the surface and bottomlayer was uniform, and if the bottomforma- 

 tion at the mouth of the fjord could affect the short waves like 

 the long waves, then the wave-series could be reconstructed by the 

 aid of harmonic analysis and with the lunar period as base for the 

 calculation. The longer the time of observations, the better would 

 be the result gained. An attempt to do this has been made and 

 failed, as was to be expected. 1 



The reason why the harmonic analysis, as usual in these cases, 

 gave an obscure result will be explained later on. The moon-waves 

 are undulations in the border-layer of the sea and subject to all 

 circumstances that affect boundary-waves. The phenomenon would 

 disappear if the seawater became homogenous. On encountering 

 submarine ridges the waves change into cascades or breakers. If the 

 ridge rises to or above the level of the border-layer as in the middle 

 and south part of the Cattegat, then the oceanic tidal wave is broken 

 by innumerable phase-ruptures just as the flood-wave of the sur- 

 face breaks when encountering an obstacle. In other words: the 

 character of the medium precludes the use of the analytic method. 



I found the insufficiency of the analytic method when trying 

 to reconstruct the sunspot curve for 150 years by the aid of the har- 

 monic analysis. I found that the sunspot curve could not be recon- 

 structed on the basis of any given period for more than 34 years. 

 Then a new periodicity set in. In reading up the subject I found 

 that Schuster before me had had the same experience with the same 

 problem. He found that no single period dominated the sunspot fre- 

 quency permanently; one period would prevail for a time and then be 

 supplanted by another. He discerned a number of such periods, 4.7 

 years, 8.34, 13.5 and finally the most important of them all, 11.125 



years. He says: »The existence of a number of definite periods 

 can not be doubted whatever we may think of their numerical rela- 

 tionship. The recurrence of the maximum activity of each period 

 seems to take place with an accuracy which may be equal to that 

 of orbital revolution, but the characteristic property of these periods 

 is the great variability of the activity*. In other words, the peri- 

 ods succeed one another. The one will attain its full activity when 

 the influence of the other abates or ceases (becomes latent). The 

 recurrence of the sunspots give an instance of periodic phenomena 

 with short terms of existence. 



The recurrence in the Saros period of occultations (the eclipse- 

 cycle) is the best defined and most lasting of the phenomena 

 which depend on the lunar periods. The coincidence of the lunar 

 periods in Saros is as follows: 



223 synodic lunar periods = 6.585,32 days f Saros: 

 242 draconitic » » = 6.585,34 » = 18 years %%% 

 239 anomalistic » = 6.585,55 » [ days 

 The tropic lunar period does not fit so well into Saros as the 

 others. Because of the otherwise high conformity the period »Saros » 

 of the eclipse-phenomenon, however, obtains a long term of existence. 

 The lunar eclipse repeats itself 48 — 49 times in regular succession 

 with a very small displacement in time and place 865% years once 

 in 223 months or 19 eclipse years (of 346.62 days); a solar eclipse 

 repeats itself 68 — 75 times till the end of its term of existence in 

 1 260 years. 



Probably the »long'evity» of the eclipse-series, known of old 

 in China and Caldea, caused the belief that all phenomena originating 

 in celestial causes possess unlimited periodicity and that every such 

 phenomenon in Nature could by the aid of the harmonic analysis 

 be theoretically reconstructed with a higher accuracy the longer 

 the chain of observations. 2 By demanding too much of permanency 

 many phenomena investigated by meteorologists and oceanographers 

 have been classed as irregular and incidental, though in reality they 

 are bound by laws and periodic although their periodicity has but a 

 brief existence. Often the apparent periodicity of a phenomenon 

 gets obscure and as if blotted out by other periods somewhat in the 

 manner as a seawave when watched for a time appears to dissolve 

 and merge into other wavesystems. Yet no one doubts that every 

 particle of water will perforin its orbital revolution with perfect 

 though very complex periodicity. 



An analysis of the variation of the tide-generating force shows 

 its period to be very complex. No less complex will its effect 

 on the sea and the atmosphere be. The ordinary way of reconstructing 

 complex periods by adding a number of sinus- functions into a Fou- 

 rier's series with constant coefficients, requires that all periodically 

 acting agents shall act continuously and uniformly through the ages. 

 This is not the case with the constellations of the sun, moon and 

 earth in their action on the photosphere and the corona of the sun and 

 on the hydrosphere and atmosphere of the earth. 



Phenomena which cannot by harmonic analysis be recognised 

 as periodic are otherwise disposed of. They are treated as incidental 

 and eliminated from the calculation by the ordinary meteorologic 

 way of elimination by averages using the method of least squares, 

 which method presupposes the phenomena to be independant of 

 one another. In this way the relation between many hydrographic 

 and meteorologic phenomena is blotted out and to Chance is accorded 

 far too important a part. The individuality of a phenomenon and 

 its relation to similar phenomena are both underrated by allowing it 

 to be represented by an average which is the exact expressive of 

 a situation unknown to Nature. 



Both the harmonic analysis and the calculation by averages 

 are used quite indiscriminately in Hydrography. I do not mean 

 that they should be left out in any case where they are of real 

 advantage but I wish to point out that they are not infallible. 

 The harmonic analysis was incapable of showing the relationship 

 between the undulations in the border-layer of the sea and the 

 lunar periods when analysing the 4% years series of daily observa- 

 tions on the basis of the same lunar period. Yet the relationship is 

 quite evident in the diagrams which depict the real situation from 

 the daily observations put together. 



In the Swedish oceanographic work, of which G. Ekman and I 

 are the leaders, we have sought to collect such snapshots » of the 

 situation in the sea at different seasons in a succession of years in 

 order to study the periodicity of the oceanic circulation and its 

 causes. This too was the object of the joint investigation of the 

 Swedish and Danish Commissions in 1891 — 1899 carried out by sea- 

 sonal cruises 4 times annually which gave a general view of the 



1 A similar instance is mentioned in my paper: The connection between hydrographical and meteorological phenomena. Quart. Journ, R. Meteorol. Soc. July 1912. 



2 On the other side it is obvious that the harmonic analysis must comprise a number of periods — the more the better. For a shortlived or cyclic perio- 

 dic phenomenon — - as the moonwaves, the sunspots a. O. — it consequently must give vague and indistinct results if indiscriminately employed. It seems pos- 

 sible that a number of meteorologic phenomena (as f. inst. the variations of high and low pressure, of dry and wet periods etc.) are in reality periodic and of cos- 

 mic origin although their real nature is obscured because the medium (the ocean & the atmosphere) is subject to alterations (of stratification and otherwise, etc.) 



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