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CHAPTER II TIDAL THEORY A. T. DOODSON Liverpool Observatory trod Tidal Institute The ultimate state of knowledge of the tides will exist when the tides can be explained quantitatively without reference to observation. Such an ideal is still very far from attainment, though steady progress is being definitely made towards it. So far as the forces which give rise to tides are concerned there is no difficulty in stating these with extremely great accuracy, since the distances, masses, and general motions of the sun and moon are well determined and the mathematical expression of these forces is not complicated by the physical properties of the particle, nor by its relations to other particles Which the forces act. It is not very easy,: though, without recourse to mathematical formula or compli- cated 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 perio- dicity is naturally somewhat complex, and under such circumstances the expressions for the forces are best resolved into a large number of simply- periodic terms, proportional to cosines of angles which increase uniformly with the time. Also, it is usual to deal with the "potential" of the forces rather than with the forces themselves-the potential, of course, is related to the work done against the forces by bringing the particle from infinity to its present position. The actual forces in any specified directions are derived from the potential by ordinary processes of difEer- entiation in the required directions. Such expansions have been given, in increasing detail and accuracy, by Kelvin, Ferret, Darwin, and Doodson; the last-mentioned expansion is more than sufficient for all preseIlt demands of the subject. * For explanations of tides in popular language, see textbooks by Darwin and Marmer; the latter is probably the best textbook extant. 19
20 FIGURE OF THE EAR TH The peculiar advantage o:l such expansions is really associated Title a principle formulated by Wallace, who stated that any periodic variation in the forces must inevitably cause in the tide a variation with exactly the same period. All modern methods of tidal analysis are based upon this principle. The result is that we can consider separately the various terms of the potential, and, to a first approximation at least, we can consider their effects in the tide as independent. It is usual, therefore, in mathematical investigations to consider the effects of a simply-varyin<, f'oree, represented by a cosine term whose phase increases with the time. The expansion for the tide-genera/in<, potential indicates that the principal terms have periods of about hall' a solar day; other terms, in decreasing importance, have periods of about one day, a third of a day, and a quarter of a day. Consequently, we expect tidal variations to he resolvable into terms having similar charaeteristies, and the potential gives us a good idea of the more important terms to be looked for. Indeed, from it, we can readily deduce the main eharaeteristies of tidal motion. This development, also, is used as a standard of reference for tidal analysis; the actual amplitude of the tidal term is usually stated but the phase is related to the phase of the corresponding term of' the potential by a constant appropriate to the place (the phase-la,; the amplitude and phase-lag, as determined by analysis, are the " harmonic constants " for the term in question. Several terms have " speeds " (rates of' movements of phase) so nearly equal that it is found desirable to combine them into one constituent whose amplitude and phase are the amplitude and phase of the pre- dominant term but modified by slowly varying quantities which can be considered as constants within a period of' about a year. This is per- missible 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) these l:'orces, 2) gravity, varying, with position, 3) the rotation of the earth and 4) intrinsic -i'orces characteristic of the fluid thus the particle suffers pressure from other particles, it may expand * See paper by Doodson, Perturbations of harmonic tidal constants. Proc. Roy. Soc. A. 106: 513 ( 1924) .
TIDAL THEORY 21 or contract, it suffers friction with other particles, and so forth. Fortu- nately, we can assume water to be incompressible, with subsequent simpli- fication of the problem. Now these forces can be expressed mathematically with great exactitude so that differential equations can be formulated; normally, there would be three dynamical equations of motion required, on account of possible motions in three independent directions, together with an equation, called the equation of continuity, which expresses the invariability of fluid volume within a given space. Such equations were first given by Laplace and they have been critically examined by Lamb and others. The exact equations, however, are seldom used by mathe- ~naticians because of the great mathematical difficulties involved; the subject has developed, as is usual, by easy stages involving the assumption o:t' conditions which are not always even rough approximations to those actually existing. Fortunately, the equations of motion can be simplified very considerably, especially if the vertical motion is known to be small relatively to the depth; this is always the case except in estuaries and shallow bays, whose influence on the main oscillations in the large oceans must necessarily be small; even in these apparent exceptions the simplified equations, on a priors grounds alone, are quite good approximations. Thus we can usually ignore in the equations of motion all terms (includ- ing, of course, frictional terms) depending upon the squares and products of small quantities. Further, since the vertical acceleration is small compared with the horizontal accelerations we can neglect it, in conse- quence 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. The importance of these free oscilla- tions consists in the liability of the tidal oscillations to become larvae if the period of any constituent of the impressed force is nearly equal to that of a free period. In case of exact agreement of periods it is necessary v . . ~ , . . ~
22 FI GURE OF THE EAR TH to rely on friction to prevent infinitely large oscillations, so that the approximate equations of motion are no longer appropriate, because :frietion has been neglected and vertiea.1 motion has been supposed to be small compared with horizontal motion. Our knowledge of tidal motion over the earth is su:Fieient to show, however, that it is only in a few cases that such resonance is su~eiently great to require us to abandon the use of the approximate equations of motion. The application of these equations to tidal motions in specified basins, with the earth rotating as in actuality, is very difficult and only in a few cases has it been possible to obtain solutions of the problems, some of which we shall proceed to discuss. The most famous of these solutions yields the "equilibrium tide," so called because the resulting spheroid of liquid, when the fluid covers the whole earth, is the equilibrium form that the free surface of the water could take if the moon remained over one meridian of the earth. To obtain this solution, under actual conditions of motion of earth and. moon, it is necessary to assume that the water loses its inertia while it retains its gravitational properties. The resulting free surface is a " level surface " under the combined action of the earth's gravity, centrifugal force, and the disturbing forces; on this surface the sum of the potentials of the forces must be constant, for no work would be done on a particle in moving it along the surface. If g is the height of tide, g the acceleration due to gravity (supposed constant over the earth for all tidal purposes), 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. It can be shown that if a harmonic term of V/g is E cos +, where E is constant and ~ is. a function of the time, then the eorresponding.term of V/g-C is qE eos (+-cry where q and a are constants depending on the basin and varying from place to place. Now this result will be shown to be of very great im- portanee in the exposition of the subject of tidal theory, for we are intimately concerned with the effects of boundaries upon tidal motion; the "eorreetion," as it is called, may have an important influecee on the time of high water, which is no longer synchronous with the time of maximum of the potential. The corrections have been evaluated by Darwin and Turner for the equilibrium tide as a whole, using an approximate representation of the
TIDAL THEORY Y 23 land surface of the globe; they found that the corrections were of no importance for the oceans as a whole, as we know them. Possibly a much more important correction to the equilibrium tide is that which takes account of the self-attraction of the tidal spheroid; this has been evalu- ated ~ and it appears that the expressions for the tides are thereby in- crea.sed in the ratio 1 12. It is extremely difficult, however, to correct the equilibrium tide for both land.a.nd self-attraction simultaneously. The equilibrium theory has received a degree of attention out of pro- portion to its merits, simply because of the facility with which it is obtained. No results can be deduced from it, in general, which cannot be deduced from the attractive forces (or their potential) with equal ease and with greater generality. However, its present importance is that it is a convenient standard of reference in tidal matters; the contri- butions 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. In small basks, however, it is reasonable to suppose that we can neglect the motion of the water. The dynamical equations of motion describe the relationship between the components of current on the one part and the difference between the actual tide and the equilibrium tide on the other. If the water is deep then the currents must in any case be small; also, the zero values of currents transverse to the boundaries must also tend to keep down the strengths of the currents. Hence it appears that in small seas we may expect that the actual tide will approximate to the equilibrium tide, suitably corrected it should be noted that this expecta- tion has not been rigidly proved. NVe have also to determine the possible effects of resonance, and simple calculations give the following free periods for a non-rotatin~, rectangular basin: Depth 100 feet Length = 100 miles 5.2 hours Length = 500 miles 25.9 hours 1~000 feet 1.6 hours 8.2 hours 10,000 feet 0.5 hours 2.6 hours These will serve as a useful, though very rough, guide to the free periods to be expected. It is clear that for a basin 500 miles long the mean depth must be somewhat greater than i,000 feet before we can feel reasonably sure that marked resonance of the semidiurnal tide is * See Thomson and Tait, Natural philosophy, 2nd ea., Pt. II, p. 815.
24 FIGURE OF THE EARTH unlikely to occur. Supposing resonance to be absent, then the general result sketched out above is applicable to any shape of small sea and to any law of depth provided that the mean depth is not small, and the problem is simply to determine the correction, C, referred to above. For a rectangular basin it is a simple matter to show that the small portion of the equilibrium spheroid may be taken as a plane and that the condition for continuity of volume yields the result that this plane must always oscillate about its central point. The results are suf~cient.ly interesting and important to justify the formula for the motion being quoted; if we take axes Ox, Oy eastwards along the parallel of latitude and northwards along the meridian respectively, with A as the latitude at the central point O; ~ the speed of the tidal constituent,; t the time, measured from the moment of maximum tide-~,enerating potential at O; then the elevation is given by I=-A cos A(y sin A cos crt+x sin at) where A is a positive constant. The formula does not hold near the Pole. When t=0 we have high water at all points south of 0, and low water at all points north of 0, on the meridian. Three hours later, ~rt=,r/2, we have high water along Ox to the west of 0, and low water east of 0. 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. A line joining points at which high water is being experienced simultaneously is called a cordial line; it is not usually a straight line. It is very con- venient to describe the tidal motion in any basin by means of cotidal lines. In the present instance the cotidal lines all radiate from one point at which no tide is ever. experienced; such a system is called an amph~- drom~c system and the no-tide point is called an amphidrom~c point. Again, ire this instance, the times of' high water are progressively late as we move around the a.mphidromic point in the negative sense (that. of the hands ol' a watch) and the rotation is said to be negative; in the southern hemisphere the rotation is reversed. It is impossible to over- est.imate the importance which we must attach to amphidromic systems such as we have described, for such systems are the rule rather than the exception, not only in small seas but also in the large oceans. Further interesting, and typical, conditions can be deduced from the simple formula for the rectangular basin; the cotida,1 lines are not, equally spaced out but spread out from the center lily two fans symmetrically Placed, the fans being almost closed near the equator; in the limit we have the tidal motion near the equator (A=O) given by I=-Ax sin at
TIDAL THEOR Y 20 so that high water occurs simultaneously throughout the western half, and low water simultaneously throughout the eastern half, when at=~/2. We thus have two cot~dal areas and the water oscillates about the central meridian in a see-saw fashion; at t=0 the whole surface is horizontal. Such cotidal areas are also experienced in the oceans. The formula quoted above is true for any small flat sea, if O is the centroid in the mathematical sense, and similar results have been obtained for a circular and also for a serr~icircular sea. In the latter case an exact solution has been obtained by J. ProuUman and this indicates that for a sea comparable with the Black Sea the solution is only just appreciably different from the corrected equilibrium solution. 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. Also, the rotational effects of the earth are usually negligible for a narrow bay; the rotation of the earth, of course, tends to throw the Hater to the right o:t' its motion, but the result is inappreciable except in a broad basin. A number of simple cases have been worked out, such as a) depth constant, breadth varying with distance from the closed end; b) breadth constant, depth varying with distance from the closed end; c) breadth and depth varying with distance from the closed end. These solutions show that tides tend to increase in range towards the closed end of' the bay, owing to the restriction of the cross sectional area. Ultimately, frictional effects become prominent, but it is remarkable how these simple approximations account for the la.r'~,e tide in an estuary such as the Bristol Channel. Considering now the subject of' broad seas which carrot be considered small enough for the corrected equilibrium theory to be applicable, then, even when rotation is absent, the problem is a difficult one. Certain cases have been evaluated on the supposition that the earth does not revolve and the fluid is bounded by two meridians; the solutions for these special boundaries show an intricate series of amphidromic points. In order to gain some conception of the tidal motion in enclosed oceans a considerable degree of attention has been paid to motion in long canals; thus G. B. Airy in 1846 discussed the case of' a canal of varying, rectan~u- lar section lying, alone, any complete great or small circle of the earth's surface, and his most important cases were treated by Lamb (1895~. In 1910 H. Poi~:~care discussed the general principles -for the treatment
26 FIGURE OF THE EARTH of a network of canals, and H. Lamb and L. Swain in 1915 elaborated the solution for a canal of uniform rectangular section occupying only a portion of the earth's equator. Until quite recently no solution was available for a large ocean except that given by Laplace in 1776 for an ocean of constant depth covering the whole earth, or bounded only by the equator. This solution has given rise to much discussion, as it was disputed by Airy and by W. Ferret, but was vindicated by W. Thomson (Lord Kelvin) 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. In the tidal problem a large number of functions can be expressed, either algebraically or numerically which formally satisfy the equations, and these functions have to' be appropriately combined to satisfy the conditions of zero velocity trans- versely to the boundaries. Returning to Laplace's problem, there is only one " boundary-eondi- tion '' and that is at the equator; considerations of symmetry show that the velocity transverse to the equator must be zero. Now Laplaee, without adequate explanation, essentially combined a particular integral and a complementary function so as to satisfy this boundary-condition in one operation. Thomson's explanation, to many, only made the matter more mysterious than it really is, but the method of combining two separate solutions, as indicated, is very simple and the full solution has been elaborated in this way by Doodson (1928) for the semidiurnal tide. Laplaee's solution has been elaborated by Hough, who took also into account the self-attraction of the water. The results obtained by Laplace and Hough may be summarized as follows: R.\T:IO OF TIl)E lo CORRESPON1)IN(! EQLT1LIBRIUM TIDE On the Equator Depth of ocean f. Long-period Laplace 0.455 tides 1 Hough 0.426 '`,SGO foot 0.551 14 r,)0 feet 29,040 feet .SS.0£0 feet 0.708 0.681 0.817 0.796 Semi-diurnal I Laplace 7.4341.821 11.259 1.924 tides 1 Hough 7.955-1.502 234.9 2.139 At the J Long-period (Laplace 0.154 Pole :` tides Though 0.140 0.266 0.470 0.443 0.651 0.628
TIDAL THEOR Y 27 Also, at the Pole, the semicliurllal tides are practically Vilely by the Equilibrium Tide, but in any case the t:icles there are smalls The diurnal ticl.es are rather remarkable, in that their ranges are everywhere very small indeecl. On the whole, the lon~-period tides are " direct " ill that the:: have hi~,h-~-ater s:-nchrol~ousl:- faith tl-~e equilib- rinn~ titles; the equilibrium tide has noclal lines (lines o:t zero range of tide) and these are slightly shifted in the solutions obtainecl by Palace v ,, , ~ 4 ,} ,) AT ~ ~1 · ~ · ~ ~ · ~ · ~ I · I ancl llou~h. 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) and for larger depths they are " clirect." A seconcl critical depth is indicated below 7,260 feet. By allowing -for the mutual attraction o:t the riveter llou~,h obtained slightly clifferent values for the critical depths; also, I-Iou~h -iou~cl that when the depths are 29.04-0 feet and rY,260 feet then the periods of free oscillations are i? hours ~ minute, and 12 hours ~ minutes, respectively, anal he concluded that the tides will not become excessively large unless there is very close agreement between the periods o:t the tidal constituent and free oscillation. The case of a POINTS Ocean iS readily treated in the sane manner as that of the ocean bounded by: the equator, and a solution has been pub- lished be G. It. Golclsboro~,l:~ (~)13; see also Doc~aso~' 1998~. His results Cain be s~narizecl as -follows: ~. , . . ~. . . . RATIO OF TIDE IN POLAR OCEANS-, BOUNDED NT LATITUDE 60, TO EQUTI IBRTUN! TIDE At the pole At the boundary: ~, ~- Lo7~-period Lolls -per iod Diurnal Semi-di~rnal tides tides tides tides .. 0.100 - 0.216 - 2.86 -1.0~0 . 0.133 - 0.2a=7 - 1.16 1.030 . 0.158 - 0.269 - 0.194 1.014 . 0.176 - 0.2~6 -~.386 1.003 ]]el)tl, ~ 260 feet ........ 14.520 feet 29.040 feet 58,080 feet For the lon~-period ticles, if He c onsiclered the ocean as s~nall and computed the correc tion as explainecl previously, the ratio of tide to equilibrium-tide would be 0.192 at the pole and-0.293 at the bounclary, so that if the depth is very great the corrected equilibrium-tide is a close approximation. The se~ni-diurnal tides are practically equal to the equil:ibriu~-ticles of the same species, the " correction " being, zero. The diurnal tides are of most interest in. view of the fact that these are very- small incleed when 3
28 FI CURE OF THE EAR TH the water covers the whole earth; we have a further illustration here of the remarkable effects of boundaries. Goldsborough finds that the largest critical depth for resonance of the semi-diurnal tide is about 526 feet when the boundary is in latitude 60° and about 38 feet when the boundary is in latitude 7~° 3O'; the smallness of these depths precludes the possibility of resonance in Polar oceans of depths comparable with actual average depths of the oceans. An ocean bounded by two parallels of latitude can also be similarly discussed. The results obtained by Goldsborough (1914) 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. The effects of the boundaries on the diurnal tides may again be noted, extremely large values being ob- t.ained in one instance. Goldsborough determined the critical depths for semidiurnal tides and for diurnal tides. The solution of the problem of tides in an ocean e~tendq~r~g front Pole to Pole arid bounded by two meridians is much more di~eult than any yet considered, for it is theoretically necessary to deal with an infinite number of functions in order to make the velocity zero a.eross the equator (for the semidiurnal tides) and across the bounding meridians. The
TIDAL THEORY 2'3 problem can be formally solved but the actual arithmetical solution is exceedingly dif-Dcult. A solution has recently been published by Golds- borough for ail ocean whose depth varies as the square of the cosine of the latitude, in which case the tidal equations are somewhat simplified. The form of the solution does not permit of the determination of the free periods of oscillation but it gives, what is almost as useful, the critical depth at which resonance takes place. By taking the bounding meridians 60° apart and the mean depth as 16,980 feet, results roughly applicable to the Atlantic Ocean were obtained. Previous papers by the same author had demonstrated that large semidiurnal tides cannot be generated in the polar oceans with the depths known to occur. I-Ie now remarks that this points conclusively to tile impossibility of large semi- diurna.1 tides in the Southern Ocean, even though some allowance be made for the Antarctic Continent. Consequently, large oscillations in the Atlantic Ocean must be~ due to resonance and not to propagation northwards. His results indicate that the critical depth. in the cited case is 1~o,500 feet; the mean depth of the Atlantic Ocean is 12,?00 feet and for resonance to occur with this mean depth the bounding meridians should be about 53° apart. 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. Shell further elaborated, and fully illustrated, this solution should give a fairly satisfactory, and extremely instructive, representation of the tidal motion in such an ocean as the Atlantic Ocean. The theoretical studies outlined in the preceding paragraphs are ex- tremely valuable in spite of their artificiality, but the important problems of the future will be concerned with coccal seas arid oceans. The ideal solution is one which tallies account of all irregularities of form of basin, and only in a fear instances has it been found possible to bring all the irregularities of a natural body of water, just as they exist, into a dynamical treatment. Apart from the investigation of the periods and elevation-form of seiches, which are essentially free oscillations, in Lal~e Geneva, the only instances of complete treatment in this respect are due to Sterneck and Decant, who have dealt with narrow seas and. channels such as the Adriatic Sea and the lied Sea. These examples deal with forced oscillations, the motions at the mouths of the seas being taken from observations. Such methocls, however, have been applied to seas and channels so narrow that transverse motions have been negligible, and the incorporation of all geographical details for a broad sea such as the North Sea has not yet been achieved. In general, therefore, the de
30 FI CURE OF THE EA R TH tailed explanation of tides is very incomplete, and attention has been largely `~,;ven in the past to the <<reo<`rraphical distribution, usually cle- scriDecl by coticlal lines, and ut.ili.~;n~¢, actual observations on the coasts. It will be recalled that a cotiflal line connects points at which high water is svnch.ronous; briefly speal~i~,, each tidal constituent has its own set of coticlal lines, and commonly they are drawn either for the principal lunar const;.tuer~t or-for the conjoin." principal lunar and solar constitu- ents, conspiring at.-.fu.ll and chase of tl~e moon,--for the representation of semicliurr~al titles. For cliur~:~al tides the two principal diurnal con- stituents are combined together. The best methocl, and the only one readily interpreted dynamically, is tl~at which clears only with a si.nt,le constituent. The first exponent oft' the art of' drawing, coticlal lines was NYhewell, who contented himself at first with attempts to picture the notion near the coasts. Ike enunciated a theory of a progressive wave, and a remark- able fact is that the phases of the principal lunar semi.cliurnal title. M appear to i.nclicate such a progression northward from the Southern Pacific Ocean, and also the principal solar constituent S., has a phase relationship with 31., which appears to support such a theory; the phase clifference, on such a theory, should. steadily increase northwarcl.s by about ~ · .. .. 1 ~ Per hour n,acl the rI~-~ro~ its 3~r~ is ~11~8 the. " age." It should ~ ~ ~ · ~ ~ ~ ~ ~ ~4 ~ ~ g~ vat ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ e) ~ be noted that TVhewell was much more cautious than his successors, whose efforts has-e persisted in spite of the serious criticisms launched at the theory. Consid.eratio~s of' the transport of' ener~,::, as pointed out by Poincare, preclude tl~e possibility of' a. pro~,ressive wave northward. One of' TVhewell's last efforts for the Forth Sea is illu.stratecl in Figure 1; the lines are drawn on the supposition of' a progressive wave whose rate of' travel depends upon the clepth; later exponents of this theory have evolved maps with lines showing, much more curvature than NVhewell's lines. The amphidromic point in the Flemish Bight should be noticed, as it led to Captain Hewitt's takings observations for its verification. 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. hi:. Ferret, for one, disbelieved it, and went so far as to say that the North Atlantic tides would not be much affected if' a barrier were made from ~Tort.h A:t'rica to South America. Goldsborough's recent investi`~,ations have gone far to demonstrate what ninny have hell, that the ticl.es ill the Atlantic Oceal1 are generated within the ocean, Ceils, ma~,.llified by resonance, without appreciable propagation inwards from the Antarctic Ocean.
TIDAL THEORY 31 The importance of resonance was much. emphasized by P. 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. These latter assumptions malice it ex to W.? (~e In'\ - ...: ~:'~''~\W ~ '.'...2'.''..'.'.2.'.'.''~ ~ 4'~'-~'.'.~', ~,~ 3 // ~ :: ~/ )~43/ 4'~"'~':~1: FIG. 1. Cotidal lines for the North Sea according to W. Whewell, 1836. tlemely doubtful whether Harris' hypothesis can lead to a satisfact,ory- explanat.ion of' the tides in any of' the oceans, but undoubtedly the importance which he attached to resonance has made for progress in these matters. In 1~)20 If.. Sternest proclucec~ a. set of coticlal lines for the se~idiurnal constituents for all the oceans, and in 1922 a corresponding, setf'or the ~l.iurnal constituents. The orals, principle ut:ilizecl appears to lie that co- tidal lines for phases cli~'erin~, by ~ shall be as nearly parallel as is con
32 PI GURE OF THE EAR TH sistent with observations, and there may be theoretical justification for this principle. It will be most useful at this stage to consider the tides in the North Sea. Omitting all frictional consiclerations and, -for the present, the rotation of the earth, then any tidal oscillation in the ~'\tlant.ic Ocean would maintain a "standing oscillation" in the North Sea with two 9/ '~41\ /~~ it' rat ::~ ~ .',,.,,,,.,,,,, ~ 1~\\\~\ FIG. 2. Cotidal lines for the North Sea according to R. A. Harris 1904. nodal lines about the lat.itucles oo° id. and .~8° IT, as may readily be shown 'iron considerations of' emotion in a non-rotatin~, trough. Along these lines would be zero range of' tide, but st.ron~, tidal currents across the lines would exist three hours before or after the hours of high water in acljoinin~, regions. The effect of the earth7s rotation on these currents would be a surface ~,raclient along, the line, and so there would be a tidal oscillation of level on. the line, the range now being zero at only one point of it. The phase of this superposed oscillation will thus change
TIDAL THEORY 33 b.y 180° as we pass through this point, and similarly the phase of the original standing oscillation changes by' 180° as we pass through the point transversely to the nodal line. NATe have, in fact., an amph.idromic point. An explanation of this sort agrees fairly well with Sterneck's chart. ~ purely mathematical investigation of the tides in a rot.atin~ rectangu- lar gulf of constant depth and with approximately the same dimensions W~ ..; .: . . ~ ~ ~ ~ . ~ .- .,: I, :2. :~J '-I,/ ;~ A'''.' '...,'..-2~ ~ it, FIG. 3. Cotida1 lines for the North Sea according to R. Sternecl;, 1920. as the North Sea has been ~,~iven by O-. I. Taylor, and his results show two amphidromic points s:~n~Letricallv placed in the gulf. Sternecl<'s map is thus an advance on previous maps and it gives a good idea of the actual conditions. Ii we include the effect of friction, however, it becomes apparent that the range of the tide decreases north- wards along the continental shore of the North Sea, and thus there is a bias o-t the amphiclromic point towards this coast, the bias increasing, northwarcls; the northerly amphiclromic point should thus be much nearer than the southerly point is to the coast. , ~,. ,. ~
34 FI GURE OF THE EAR TH Theoretically, the position of' an amphidromic point depends upon the speed of the tidal motion, and each harmonic constituent of the tides has its own amphiclromic Anoint, though all constituents of' one species will have their amphiclromi.c pouts clear to one another. Thus, actually, the range of tide will not always vanish, though it will be small near the amphidromic point of the principal constituent. At and near the amphi- dromic 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. These have been investigated bar Harris and by ProuUman. An island may have very remarkable effects upon the tides in its vicinity and in general cotidal lines converge on a cape and diverge into a bay. It is somewhat doubtful whether satisfactory cotidal charts for the larger oceans yet exist, as there is still a large element of uncertainty as regards the validity of' the hypotheses. Where approximately exact maps are required special methods must be used. Of course, observations of' the rise and fall of tide at a large number of' places throughout a sea 1 V will provide ample data for drawn such charts, but direct observation is very difficult and, even when practicable, is very expensive, so that extremely little valuable information has been obtained in this mariner. Observations of' tidal currents are more easily made, and the relation between current and elevation can be exactly expressed in the dynamical tidal equations rel'errecl to above. From the currents it is possible to compute the space-rate of' increase of' elevation in any direction and if' sufficient observations are available then methods old numerical integration can be applied to these =~raclient.s of elevation from coast to coast. For the North Sea such a method was carried out at the Tidal Institute, and use was also made of' much miscellaneous observation, with a satis- :t'actory measure of inte~-a=~reement. The resulting chart is illustrated in Figure 4 and a slightly different form of it was adopted by the British Admiralty. The position of the northerly amphiclromic point is a ~ . . . ~ . · . . · · 1 ~ ~ 1 ~ 1 ~ 1 1 _ _ __ _ _ _. `` ~ _ ~ ~ _~ _ an._ 1 ~ '' / 1~ ~ ~ little uncertain; it is possibly the case that It becomes Degenerate ~ final is, the actual point to which the coticlal lines converge may be inland) but, as the tides near Norway are very small, it does not matter appreciably where the point is situat.ecl. The chart is appropriate only to the principal lunar constituent M.>; the time of' high water at a given place, in mean solar hours after the transit of the moon at Greenwich, is equal, on the average, to the phase-l divided by 29. It may be noted that the method outlined Eves also co-ran~e lines; on the chart illustrated the semi-ranges O are given in centimetres.
[ recopy 35 TO this Outbox no Use bus been Ague Of the equation of continuity which expresses the elevations in terms of the sp~ce-gr~]ients oT the currents; ant Detest has given ~ chart utbising oI(y this principle. Obese FIG. 4. CotidulEDeG for the~ortbSe~uCco~ingtotbeTid~llDstitute, 1923. Condo.------- Cowl. ~ ~ gener~1 agreement betwceD his chart RD] Abut of the li4~1 IDsthUte but his lines Ace gore curve], especially Bear the coast. Since even the best current information is not very exact it ma re~son~b~ {e SUg- geste] that the use of the continuity equation, involving DUme~ic~1 di~er
36 FI CURE OF THE EAR TH entiations, is interior to the use of the dynamical equations, involving, numerical integrations. From the point of view of tidal theory, however, it is very interesting that two such dissimilar methods should give similar results. Returning, to the general problem of the construction of maps of co-tidal lines for 13road Seas and Oceans, a method has been devised by Proudman, based upon exact clynamical equations, which theoretically determines the tidal motion, at any one point of a sea, from a knowledge only of the coastal observations, ~eograpl-~ical deposits of the basin, and of the tide-generating forces. It requires the evaluation of certain func- tions over the sea and it has not yet been exploited. 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. been devised by Proudman, and, though the amount of computation involved ir1 its application is large, it is believed to be not entirely prohibitive. It is quite likely that the next great advances in tidal theory will be corrected with methods of numerical integration applied to a.ctua.1 oceans, for it is in the interpretation of the mathematical results that. the chief difficulties lie. Generally speaking,, therefore, we ma:- close our survey of the subject of tidal theory with the remark that of late years much definite progress has been made and that the general lines of development ~ ~ _ ~ ~ ~ ~ ~ { ~ 1 ~ 1 ~ of the subject are quite clear; It IS a mailer OI labor and patience to achieve the ultimate state of knowledge referred to at the outset. REFERENCES 1687. Newton, I. Philosophiae naturalis principia mathematics. 1780. Laplace, P. S. Mecanique c;eleste. 1833. Newell, V/ Essay towards a first approximation to a map of cotidal lines. Phil. Trans. Roy. Soc., p. 147-236. 1836. . On the results of an extensive system of tide observations..... Phil. Trans. Roy. Soc., p. 289-342. 1837. Green, G. On the motion of waves in a variable canal of small depth and width. Cambridge Trans. vol. 6 (Math. Papers, p. 225). 1845. Airy, G. B. Tides and wages. Encyclopacdia metropolitans. 1868 et seq. Thomson, W. Report of Committee for the purpose of promoting, the extension, improvement, and harmonic analysis of tidal observations. Brit. Assoc. Report, 38: p. 489-505. 1874. Ferrel, by. Tidal researches. Appendix to Report of U. S. Coast and Geo- detic Rev.
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