Physics of the Earth - II The Figure of the Earth Bulletin of the National Research Council (1931) / Chapter Skim
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Chapter II. Tidal Theory
Chapter II. Tidal Theory
Pages 19-39
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From page 19... ...
It is not very easy,: though, without recourse to mathematical formula or complicated diagrams to give a satisfactory notion of the way in which the tide-generating forces are determined; it is sufficient to say that these forces can be expressed for any given locality in formulae involving the masses of the Sun, Loon, and Earth; the distances of the Sun and Moon from the Earth; the astronomical latitudes and longitudes- of Sun and Moon; the geographic coordinates of the locality in question, and the time. These forces are periodic, since the positions of the attractive bodies, with respect to the particle considered, are periodic; the periodicity is naturally somewhat complex, and under such circumstances the expressions for the forces are best resolved into a large number of simplyperiodic terms, proportional to cosines of angles which increase uniformly with the time.
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From page 20... ...
This is permissible provided that we can assume the same relative amplitudes of these tidal terms as holds for the corresponding terms of the potential, and that the phase-lags of' the tidal terms so combined are identical. Experienee has shown that in general the relative amplitudes and the phase-lags change very slowly.- with the speeds of the terms, so that I'or terms whose speeds are nearly equal the assumptions made are quite probably valid; there are exceptions to the rule, however.: Having an exact kllowlecl~,e of' the external forces, the next requirement is ail exact lolowlecl~,e Ott' the hchaviour o-i' ~ single particle o-t' water under the action of' 1)
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From page 21... ...
Further, since the vertical acceleration is small compared with the horizontal accelerations we can neglect it, in consequence of which we regard the fluid pressure at any point as equal to the statical pressure due to the depth below the surface; this implies a "long wave "; that is, the wave-length is large compared with the depth, and herein is the distinction between surface oscillations and tidal oscillations. The tidal motion is a ":forced oscillation " in that it is maintained by the external forces, but if we omit from the equations of motion all reference to the external forces, then we have the equations of "free motion." Clearly, if the fluid in a limited basin is suitably disturbed and then left free, it will oscillate in a manner peculiar to the basin, and the motion will persist, in the absence of friction, with a characteristic period; there will be several periods, as a matter of fact, just as a violin string will give out a note of a definite period with overtones of periods one-half, one-third, one-quarter of the principal period; in the case of a complex basin, however, there may not be such a simple relation as this between the various periods.
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From page 22... ...
, V the potential of the external forces, then the lunar equilibrium tide = g = V/g-C, where C is a constant if the earth and moon are fixed relatively to one another One other condition remains to be fulfilled, that the volume of water has remained unchanged by the deformation. Clearly the mean value of ~ over the basin ea.n be taken as zero, so that C represents the mean value of V/g taken over the basin.
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From page 23... ...
However, its present importance is that it is a convenient standard of reference in tidal matters; the contributions of the tide-generating forces to the dynamical equations are conveniently expressed in terms of the equilibrium tide; and in theoretical investigations the results are compared with the equilibrium tide, as is also the case with tidal analysis, so far as the phases of the constituents are concerned. The failure of the equ,~t,ibr~n,m theory to express quantitatively the tides of the main oceans is due, of course, to the untenable assumption that the inertia of the water can he neglected.
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From page 24... ...
Also for any given time high water occurs simultaneously along a. straight line radiating from O; and if' this straight line is produced through O it joins all the points at which low water is then being experienced.
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From page 25... ...
The problems of oscillations in narrow bays are very varied and they are quite different in character from the problems just. discussed, for such tidal oscillations as exist are maintained by the oscillations generated in the main ocean, and the variation of the equilibrium tide within the bay is relatively negligible.
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From page 26... ...
in a very remarkable paper. Generally speaking, a system of linear differential equations can be satisfied in an infinite number of ways, and the problem is to combine these solutions so that they satisfy all the conditions at the boundaries; there is only one such combination appropriate to the given conditions but the same independent " complementary functions " can be combined in different ways with a " particular integral " to satisfy the conditions for a large number of different conditions.
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From page 27... ...
l he sem~d~urnal t:rcLcs prov:~cLe most interest; for large depths they are direct and tend to be represented by the equilibrium values as the depth increases; but when the depth is about 29,040 -feet the range of tide on the equator becomes very great. Laplace's results inclicate that a depth between 14,320; feet and 29,040 feet is " critical " in the sense that resonance occurs at that depth for squaller depths the tides are "inverted" (high water occurring, when the equilibrium tide would <-,ive low water)
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From page 28... ...
are as follows: ZONAL OCEAN BOUNDED AT LATITUDES 30 AND 14 30 : RATIO OF TIDE TO EQUILIBRIUM TIDE ..0O :30 ° T,atitude T4° 30' ~. ~ Sergei- Lon~- Semi Lon~-period Diurnaldiurnal period Diurnal diurnal Depth tides tidestides tides tides tides 7,260 feet - 1.10 - 0.100.48 0.31 - 0.5~3 0.39 14,520 feet - 1.20 - 0.39- 0.58 0.31 - 0.99 - 0 07 29,040 feet - 1 .23 - 0.84- 1.46 0.31 - 1.63 - 0.78 58,080 feet - 1.24 - 42.61.81 0.31 - 31.0 0.37 ZONAL OCE.~N BOUNDED AT LATITUDES 30 AND - 14° 30 : RATIO OF TIDE TO EQUILIBRIUM TIDE Latitude 30° [Latitude _14 ° '30' _k ~, ~ Semi- Lon~- Semi Lon~r-period Diurnal diurnal period Diurnal diurnal Depth tides tides tides tides tides tides 7,260 feet - 2.04 1.95 - 0.59 - 0.018 3.48 - 0.46 14,520 feet - 2.14 1.21 4.49 - 0~.010 2.16 - 3.85 29,040 feet - 2.20 0.45 - 1.11 - 0 004 2 19 - 0~70 58,080 feet - 2 21 - 3 00 - 6.65 - 0.002 0.69 - 3.44 Again it is found that the long-period equilibrium tide corrected for boundaries gives a close approximation to the actual tides, but for the other species of tides the corrected equilibrium tide offers no useful indication of the true tidal motion.
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From page 29... ...
Allowing for varying contours and the form of depth, this seems strong, evidence that the Atlantic semidiurnal tides are due to partial resonance; in the case Dollied out. the tides were 50 to 60 times the equilibrium tides.
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From page 30... ...
The wave theory is dying, a natural death, having held tidal theory in bondage for many years. It still -features lart,tel~ among the popular non-scientific " explanations " of' the tides.
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From page 31... ...
A Harris, and he attempted to select special portions of the water-surface of the earth in which the free tides have speeds nearly equal to that of the tide-generating constituent; he also assumed that the free tides have a general character which can be cleterm.ined without taking into account the earth's rotation, and that the reaction of the water in other parts of the oceans can be ne~lect,ed.
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From page 32... ...
2. Cotidal lines for the North Sea according to R
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From page 33... ...
An explanation of this sort agrees fairly well with Sterneck's chart. ~ purely mathematical investigation of the tides in a rot.atin~ rectangular gulf of constant depth and with approximately the same dimensions W~ ..; .: .
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From page 34... ...
At and near the amphidromic point for the principal lunar semidiurnal constituent the tides of other species may become relatively prominent. In the actual drawing of coticla1 lines much use can be made of the known effects of capes, bays and islands.
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From page 35... ...
Abut of the li4~1 IDsthUte but his lines Ace gore curve] , especially Bear the coast.
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From page 36... ...
The ideal result of the dynamical theory of the tides will be achieved when it is possible to dispense even with the coastal observations, and indeed Poincare reduced the complete clynamical explanation of the tides to a sequence of direct mathematical operations, but he fully admitted that the amount of computation required to apply his process to the actual oceanic basin was entirely prohibitive. A mathematical method of the same general nature, though entirely different in detail, has.
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From page 37... ...
Uber die Periodendauer der Eigenschwingungen des Adriatischen Meeres. Annalen der Hydrog., 39: 119-130.
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From page 38... ...
Ube~ die Einfluss der Erdrotation auf die halbtagigen Gezeiten der Adria.
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From page 39... ...
Peters, H Theorie der eintagigen Gezeiten im Sudchinesischen Meere und im Golf von Mexiko.
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