Encyclopædia Britannica, Ninth Edition/Tides/Chapter 5

V. Synthetic Method.

§ 28. On the Method and Notation.

Synthetic method.The general nature of the synthetic method has been already explained; we now propose to develop the expressions for the tide from the result as expressed in the harmonic notation. If it should be desired to make a comparison of the results of tidal observation as expressed in the synthetic method with those of the harmonic method, or the converse, or to compute a tide-table from the har monic constants by reference to the moon's transits and from the Harmo nically analysed. declinations and parallaxes of sun and moon, the analytical ex pressions of the following sections are necessary.

In chapter iv. the mean semi-range and angle of retardation or lag of any one of the tides have been denoted by H and ic. We shall here, however, require to introduce several of the H s and K S into the same expression, and they must therefore be distinguished from one another. This may in general be conveniently done by writing as a subscript letter the initial of the corresponding tide; for example H m, K m will be taken to denote the H and K of the principal lunar tide M 2 . This notation does not suit the K 2 and KI tides, and we shall therefore write H", K" for the semi-diurnal K 2, and H, K for the diurnal K a tide. These two tides proceed according to sidereal time and arise from the sun and moon jointly, and a synthesis of the two parts of each is effected in the harmonic method, although that synthesis is not explained in chapter iv. The ratio of the solar to the lunar part of the total K 2 tide is 46407; hence 683 H" is the lunar portion of the total K.J. There will be no occasion to separate the two portions of K lt and we shall retain the synthesis which is effected in the harmonic method.

§ 29. Semi-Diurnal Tides.

Mean longitude and elements replaced by hour-angle, declination, and parallax.The process adopted is to replace the mean longitudes and elements of the orbit in each term of the harmonic development of the schedules of § 23 by hour-angles, declinations, and parallaxes.

At the time t (mean solar time of port reduced to angle) let a, 5, ^ be D s R.A., declination, and hour-angle, and I D s longitude measured from the "intersection." These and other symbols when written with subscript accent are to apply to the sun. Then > being the R.A. of the intersection, we have from the right-angled spherical triangle of which the sides are I, 8, a - v the relations

tan(a-c) = cos/tan J, sin 5 = sin/sin I ......... (74).

Now's - is the D s mean longitude measured from the intersection and's -p is the mean anomaly; hence approximately

l=s- + 2esm(s-p) ..................... (75).

From (74) and (75) we have approximately

a =s+ (v - ) + 2 sin (* -p) - tan 2 7 sin 2(s - |).

Now, h being the's mean longitude, t + h is the sidereal hour-angle, and

^ = t + h-a. 

Hence

t + h - s - (v -? ) = ^ + 2e sin (s - p) - tan 2 ^/sin 2(s - ) (76).

Again, if we put

(77),

we have approximately from (74) and (75)

cos 2 5 - cos 2 A . . = cos2(s-) whence sin 2 A sin 8 cos d dS W .(78). a sin- A

Obviously A is such a declination that sin 2 A is the mean value of sin 2 5 during a lunar month. Again, if P be the ratio of the D s parallax to ner mean parallax, the equation to the ellipse described

-(P-l)=cos(s-p) 1 dP .(79). gives whence,. Ui ^, f,, . e(ff-rs) dt J

Now it appears in schedule A of 23 that the arguments of all the lunar semi-diurnal tides are of the form 2(t + h-v)2(s-%) or (s -p). It is clear, therefore, that the cosines of such angles may by the relations (76), (78), (79) be expressed in terms of hour-angles, declinations, and parallaxes. Also by means of (77) we may intro duce A in place of / in the coefficients of each term. An approxi mate formula for A is 16 -51 + 3 0< 44 cos N- 19 cos 2N. In the Report to the British Association for 1885, the details of the processes indicated are given.

Age of declinational and parallactic corrections.Before giving the formula it must be remarked that the result is expressed more succinctly by the introduction of the symbol 5 to denote the D s declination at a time earlier than that of observation by an interval which may be called the "age of the declinational inequality," and is computed from the formula tan (K" - K m )/2<r rallactic or 52^h·2 tan ("-*c m ). Similarly, it is convenient to introduce P correcto denote the value of P at a time earlier than that of observation tions. by the "age of the parallactic inequality," to be computed from tan (K m - K n)/(<r- &) or 105 h- 3 tan (/c m -/f n ). These two "ages" probably do not differ in general much from a third period, com puted from (K, - K m )/2(<r - rf}, which is called the "age of the tide."

The similar series of transformations when applied to the solar tides leads to simpler results, because A, is a constant, being 16 33, and the "ages" may be treated as zero; besides the terms depend ing on ddjdt and dPJdt are negligible. If now we denote by h 2 the height of water with reference to mean water-mark, in so far as the height is affected by the harmonic tides Mg, S 2, K^ N, L, T, R, 1 the harmonic expression is transformed into

^[1] Total semidiurnr.1 tide. COS (2V - Km) + H, COS (2V, - i C os 2 8 -cos 2 A. gQOtJ,,, 0/ . =5-7 683 H cos (2V - K ) sin- A, cos 2 8. - cos 2 A, . 01,. TT,,,,. -5-: TJ17 H cos (2V- - K ) sin- A, sin S cos 5 dS~ "683 H" _!(**-*.) ~| . J sm (2V H n Hj -] . /ftf ^^ .(81). where e is an auxiliary angle defined by H n sin /f_ - Hj sin KI tan e = ff " TT H n cos K n - Hj cos KI The first two terms are the principal tides, and the physical origin of the remaining small terms is indicated by their involving 8, 8,, dS/dt, P, P lt dP/dL The terms in dd/dt and dP/dt are generally smaller than the others. The approximation may easily be carried further. But the above is in some respects a closer approximation than the expression from which it is derived, since the hour-angles, declinations, and paral laxes necessarily involve all the lunar and solar inequalities.

§ 30. Syntheses of Solar and of Lunar Portions of the Semi-Diurnal Tide.

Commencement of synthesis.Let us write

r cos 2 A _ cos 2 S - cos 2 A OOT T,/ . * M = ., . H m + r—r -683 H cos (c* - K m ) ment 01 cos A 7 sin A ^ COS 2 A fv, x H n cos K n - Hj cos KI . e cos e 683H"sin(/c -K m ) cos 2 S - cos 2 A sin 2 A, + cos^A H n cos,c n -H,cos S . cos 2 A, ecosc sin8cos8d8r 683 H" _ . . ~1 —^A; dt Lcost*"-^)- 11 "** 11 A J cos 2 A ^dPJdtT O A I * J—L 7M C e cos 2 A, ff - -a _ H. - /c n ) " cos (KI (82).

Since observation and theory agree in showing that K" is generally very nearly equal to K,, we are justified in substituting K, for K" in the small solar declinational term of (80) involving 317 H" Then using (82) in (80),

h 2 = M cos 2( V - A 1 ) + M, cos 2( V, - /*,) (83). and one

One solar and one lunar term. If the equilibrium theory of tides were true, each H would be proportional to the corresponding term in the harmonically developed potential. This proportionality holds nearly between tides of almost the same speed; hence, using the expressions in the column of co efficients in schedule [B, i.], 23 (with the additional tide R there omitted, but having a coefficient (r,jr)^.^e l cos 4 w, found by sym metry with the lunar tide L), and introducing A; in place of u in the solar tides, we may assume the truth of the proportion

With this assumption, M y reduces to

Hence M = (84).

This is the law which we should have derived directly from the equilibrium theory, with the hypothesis that all solar semi-diurnal tides suffer nearly equal retardation. Save for meteorological influ ences, this must certainly be true.

A similar synthesis of M cannot be carried out, because the con siderable diversity of speed amongst the lunar tides makes a similar appeal to the equilibrium theory incorrect. It may be seen, how ever, that it would be more correct to write cos 2 5 instead of cos 2 A in the coefficient of the parallactic terms in M and 2/x.

The three terms of M in (82) give the height of lunar tide with its declinational and parallactic corrections, and similarly the formula for ft. in (82) gives its value and corrections.

If now T denotes the mean solar time elapsing since the moon's upper transit and y the angular velocity of the earth's rotation, it is clear that the moon's hour-angle

and, since Mcos2(V~M) is a maximum when V=M or differs from fj. by 180, it follows that ft/ (y - da/dt) is the "interval" from the moon's upper or lower transit to high water of the lunar tide. Since T is necessarily less than 12 h, we may during the interval from transit to high water take as an approximation da/dt=<r, the moon's mean motion.^[2] Hence that interval is f./(y-ff), or -fafj. hours nearly, when ft is expressed in degrees. Thus (82) for fj. gives by its first term the mean interval for the lunar tide, and by the subsequent terms the declinational and parallactic corrections.

We have said that the synthesis of M cannot be carried out as Approxiin the case of M (, but the partial synthesis below will give fairly mate good results. The proposed formula is

formula. < cos-A y sin 2 A,

• cos 2 A, (85).

These formulae have been used in the example of the computation of a tide-table given in the Admiralty Scientific Manual (1886).

§ 31. Synthesis of Lunar and Solar Semi-Diurnal Tides.

Let A be the excess of J s over O s R.A., so that

V,=V+A, and h a =Mcos2(V-M) + M,

The synthesis is then completed by writing

H cos 2(/* <p) = M + M, cos 2( A ft t H sin 2(/x - <p] = M, sin 2( A - /*, so that h 2 =Hcos2(V-</>) } -M,) J (86).

(87).

Synthesis to obtain single term.Then H is the height of the total semi-diurnal tide and (f>/(y - da/dt) or <f>l(y - ff} or fa $, when <f> is given in degrees, is the "interval " from the moon's transit to high water.

The formulae for H and 4> may be written

H = V { M 2 + M, 2 + 2MM, cos 2( A - /*, + /*)} M sin 2( A it 4- u) i- f fij^

Fortnightly inequality.They may be reduced to a form adapted for logarithmic calculation. Since A goes through its period in a lunation, it follows that H and <j> have inequalities with a period of half a lunation. These are called the "fortnightly or semi-menstrual inequalities" in the height and interval.

Spring tide obviously occurs when A =/*,-/*. Since the mean value of A is's - h (the difference of the mean longitudes), and since the mean values of fj. and /*, are J/c m, %K,, it follows that the mean value of the period elapsing after full moon and change of moon up to spring tide is (K, - K m )/2(ff - ij). The association of spring tide with full and change is obvious, and a fiction has been adopted by which it is held that spring tide is generated in those configura tions of the moon and sun, but takes some time to reach the port of observation. Accordingly (K, - K m )/2(<r - 77) has been called the "age of the tide." Age of tide.The average age is about 36 hours as far as observations have yet been made. The age of the tide appears not in general to differ very much from the ages of the declinational and parallactic inequalities.

In computing a tide-table it is found practically convenient not to use A, which is the difference of R.A. s at the unknown time of high water, but to refer the tide to A, the difference of R.A. s at the time of the moon's transit. It is clear that A is the apparent time of the moon's transit reduced to angle at 15 per hour. We have already remarked that <j>/(y - da/dt) is the interval from transit to ligh water, and hence at high water

da/dt da. fdt A = A -1 3 r~. L <j> (89). y - da/ at

As an approximation we may attribute to all the quantities in Referhe second term their mean values, and we then have

ence to i . . ff-fl.. moon's transit. and y-a y-ff (90).

This approximate formula (90) may be used in computing from (88) the fortnightly inequality in the "height" and "interval."

In this investigation we have supposed that the declinational and tarallactic corrections are applied to the lunar and solar tides before their synthesis; but it is obvious that the process might be eversed, and that we may form a table of the fortnightly inequality jased on mean values H m and H,, and afterwards apply corrections, riiis is the process usually adopted, but it is less exact. The labour )f computing the fortnightly inequality, especially by graphical nethods, is not great, and the plan here suggested seems preferable.

§ 32. Diurnal Tides.

Diurnal tides not easily treated synthetica;;y.These tides have not been usually treated with completeness in the synthetic method. In the tide-tables of the British Admiralty we find that the tides at some ports are "affected by diurnal inequality"; such a statement may be interpreted as meaning that the tides are not to be predicted by the information given in the so-called tide-table. The diurnal tides are indeed complex, and do not lend themselves easily to a complete synthesis. In the harmonic notation the three important tides are Kj,0,P, and the lunar portion of Kj is nearly equal to in height, whilst the solar portion is nearly equal to P. A complete synthesis may be carried out on the lines adopted in treating the semi-diurnal tides, but the advantage of the plan is lost in consequence of large oscillations of the amplitude through the value zero, so that the tide is often represented by a negative quantity multiplied by a circular function. It is best, then, only to attempt a partial synthesis, and to admit the existence of two diurnal tides.

We see from schedules [A, ii.] and [B, L], 23, that the principal diurnal tides are those lettered 0, P, Kj. Of these K x occurs both for the moon and the sun. The synthesis of the two parts of Kj is effected without difficulty, and the result is a formula for the total K! tide like that in [A, ii.], but with the v which occurs in the argument replaced by a different angle denoted as v . If, then, we write

Partial synthesis. the three tides 0,1^, P are written as follows: =f H cos(V -/c ), K 1 = f H cos(V -ic ), P =-~R p cos[ -K -C2h-v ) + K -K p )] ...... (92).

The last two tides have very nearly the same speed, so that we may assume K = K P, and that Hp has the same ratio to H as in the equilibrium theory. Now, in schedules [A, ii.], [B, ii.], 23, the coefficient of KI, viz., H (the sum of the lunar and solar parts), is 26522, and the coefficient of P, viz., H^,, is 08775, so that H = 3 023 Hp, or say = 3Hp. Hence we have

x + P = H [f - cos (2A - v )] cos ( V - ic ) - H sin(2A - v ) sin (V - K ). If, therefore, we put R cos ^ = H [f - cos (2A _ _ _ _, y /)] } 03), Kj + P = R cos ( V + ^ - K ).

It is clear that ^ and R have a semi-annual inequality, and there fore for several weeks together R and ^ may be treated as constant. Now suppose that we compute V and V at the epoch that is, at the initial noon of the period during which we wish to predict the tides and with these values put

f = K O - V at epoch, f = K - V at epoch - ^.

Then the speed of V is y-2<r, or 13 94303 per hour, or 360 - 25 3673 per day; and the speed of V is 7, or 15-0410686 per hour, or 360 9856 per day. Hence, if t be the mean solar time on the (7i+l)th day since the initial moment or epoch,

Y - KO = 360n + 13-943 t - f - 25 367n, Y + ^ - K = 360?i + 15-041 t - f + 0-986?i.

Diurnal Therefore the diurnal tides at time t of the (n + l)th day are given by

= f H cos[13 I> -943t-f -25 -3677i] (94) | correc| tions to } height of andL.W. > =R cos[15 -041t-f + 0-9867i]

If we substitute for t the time of high or low water as computed simply from the semi-diurnal tide, it is clear that the sum of these two expressions will give the diurnal correction for height of tide at high or low water, provided the diurnal tides are not very large. If we consider the maximum of a function

A cos 2(t - a) + B cos w(t - /3),

where B is small compared with A and n is nearly unity, we see that the time of maximum is given approximately by t = a, with a correction 5t determined from

- 2A sin (25t) - nB sin n(a - /3) = 180?iB or 5t= .. Diurnal correc tion to time of H.W. andL.W. In this way we find that the corrections to the time of high water from and K x + P are 5t n =- 0^988(1 - sin [13 943 1 - ft - 25 367n] (95), 5t = -O h 98s(l+? gsin[15 -041t-f + 0-986n]

H denoting the height and t the time of high water as computed from the semi-diurnal tide. If t next denotes the time of low water the same corrections with opposite sign give the corrections for low water.

If the diurnal tides are large a second approximation will be necessary. These formulae have been used in computing a tide-table in the example given in the Admiralty Scientific Manual (1886).

§ 33. Explanation of Tidal Terms in common use; Datum Levels.

Tidal terms explained.The mean height at spring tide between high and low water is called the spring rise, and is equal to 2(H m + H,). The height between mean high-water mark of neap tide and mean low-water mark at spring tide is called the neap rise, and is equal to 2H TO . The mean height at neap tide between high and low water is called the neap range; this is equal to 2(H m - H,). Neap range is usually about one-third of spring range. The mean period between full or change of moon and spring tide is called the age of the tide; this is equal to (K, - K m )/2(ff - if), or, if K, - K m be expressed in degrees, O h 984 x (K, - K m ); K, - K m is commonly about 36, and the age about 36 h . The period elapsing from the moon's upper or lower transit until it is high water is called the interval or the lunitidal interval. The interval at full moon or change of moon is called the establishment of the port or the vulgar establishment. The interval at spring tide is called the corrected or mean establishment.

The mean establishment may be found from the vulgar establish ment by means of the spring and neap rise and the age of the tide, as follows.

Let a be the age of the tide reduced to angle at the rate of 1 016 to the hour. Then the mean establishment in hours is equal to the vulgar establishment in hours, diminished by a period expressed in hours numerically equal to,V f * ne angle whose tangent is Hj sin a/(H m + H, cos a), expressed in degrees. Also H,/H m is equal to the ratio of the excess of spring rise over neap rise to neap rise. The French have called a quantity which appears to be identical with H m + H p or half the spring rise, the unit of height, and then define the height of any other tide by a tidal coefficient.^[3]

Admiralty datum.The practice of the British Admiralty is to refer their soundings and tide-tables to "mean low-water mark of ordinary spring tides." This datum is found by taking the mean of the low-water marks of such observations at spring tide as are available, or, if the observations are very extensive, by excluding from the mean such spring tides as appear to be abnormal, owing to the largeness of the moon's parallax at the time or any other cause. The Admiralty datum is not, then, susceptible of exact scientific definition; but, when it has once been fixed with reference to a bench mark ashore, it is expedient to adhere to it, by whatever process it was first fixed.

Indian datum.It is now proposed to adopt for any new Indian tidal stations a low-water datum for the tide-table to be called "Indian low-water mark,"^[4] and to be defined as H m + H, + H + H below mean- water level. Although such a datum is not chosen from any precise scientific considerations, it is susceptible of exact definition, is low enough to exclude almost all negative entries from the table (a sine qua non for a good datum), and will differ but little from the Admiralty datum, however that may be determined. A valuable list of datum levels is given by Mr J. Shoolbred in a Report to the British Association in 1879.

§ 34. On the Reduction of Observations of High and Low Water.^[5]

A continuous register of the tide or observation at fixed intervals Observaof time, such as each hour, is certainly the best; but for the tions of adequate use of such a record some plan analogous to harmonic H.W. analysis is necessary. Observations of high and low water only andL.W. have, at least until recently, been more usual. Some care has to be taken with respect to these observations, for about high and low water an irregularity in the rise and fall becomes very noticeable, especially if the place of observation is badly chosen.^[6] Observa tions should therefore be taken every five or ten minutes for half an hour or an hour, embracing the time of high and low water. The time and height of high and low water should then be found by plotting down a curve of heights, and by taking as the true tide-curve a line which presents a sweeping curvature and smoothes away the minor irregularities. A similar but less elaborate process would render hourly observations more perfect. In the reduction the immediate object is to connect the times and heights of high and low water with the moon's transits by means of the establish ment, age, and fortnightly inequality in the interval and height. The reference of the tide -to the establishment is not, however, scientifically desirable, and it is better to determine the mean establishment, which is the mean interval from the moon's transit to high water at spring tide, and the age of the tide, which is the mean period from full moon and change of moon to spring tide.

Graphicl treatment.For these purposes the observations may be conveniently treated graphically.^[7] An equally divided horizontal scale is taken to represent the twelve hours of the clock of civil time, regulated to the time of the port, or—more accurately—arranged always to show Graphical determination of establishment, &c.apparent time by being fast or slow by the equation of time; this time-scale represents the time-of-clock of the moon's transit, either upper or lower. The scale is perhaps most conveniently arranged ofestab- in the order V, VI, .... XII, I ... IIII. Then each interval lishment, of time from transit to high water is set off as an ordinate above &c. the corresponding time-of-clock of the moon's transit. A sweeping curve is drawn nearly through the tops of the ordinates, so as to cut off minor irregularities. Next along the same ordinates are set off lengths corresponding to the height of water at each high water. A second similar figure may be made for the interval and height at low water.^[8] In the curve of high-water intervals the ordinate corresponding to XII is the establishment, since it gives the time of high water at full moon and change of moon. That ordinate of high- water intervals which is coincident with the greatest ordinate of high-water heights gives the mean establishment. Since the moon's transit falls about fifty minutes later on each day, in setting off a fortnight's observations there will be about five days for each four times-of-clock of the upper transit. Hence in these figures we may regard each division of the time-scale I to II, II to 111, &c., as representing twenty-five hours instead of oue hour. Then the distance from the greatest ordinate of high- water heights to XII is called the age of the tide. From these two figures the times and heights of high and low water may in general be predicted with fair approximation. We find the time-of-clock of the moon's upper or lower transit on the day, correct by the equation of time, read off the corresponding heights of high and low water from the figures, and the intervals being also read off are added to the time of the moon's transit and give the times of high and low water. At all ports there is, however, an irregularity of heights and intervals between successive tides, and in consequence of this the curves pre sent more or less of a zigzag appearance. Where the zigzag is perceptible to the eye, the curves must be smoothed by drawing them so as to bisect the zigzags, because these diurnal inequalities will not present themselves similarly in the future. When, as in some equatorial ports, the diurnal tides are large, this method of tidal prediction fails. This method of working out observations of high and low water was not the earliest. In the Mecanique Celeste, bks. i and v., Laplace treats a large mass of tidal observations by dividing them into classes depending on the configurations of the tide-generating bodies. Thus he separates the two syzygial tides at full moon and change of moon and divides them into equinoctial and solstitial tides. He takes into consideration the tides of several days embracing these configurations. He goes through the tides at quadratures on the same general plan. The effects of declination and parallax and the diurnal inequalities are similarly treated. Lubbock (Phil. P.M. Graphi cal pre diction. Methods of La place, Lubbock, Whewell. FIG. 3. Tide-curve for Bombay from the beginning of the civil year 1884 to the midnight ending Jan. 14, 1884, or from 12h Dec. 31, 1883 to 12h Jan. 14, 1884, astronomical time. Zero of Trans., 1831 sq.) improved the method of Laplace by taking into account all the observed tides, and not merely those appertaining to certain configurations. He divided the observations into a number of classes. First, the tides are separated into parcels, one for each month; then each parcel is sorted according to the hour of the moon's transit. Another classification is made according to declination; another according to parallax; and a last for the diurnal inequalities. This plan was followed in treating the tides of London, Brest, St Helena, Plymouth, Portsmouth, and Sheerness. Whewell (Phil. Trans., 1834 sq.) did much to reduce Lubbock's results to a mathematical form, and made a highly important advance by the introduction of graphical methods by means of curves. The method explained above is due to him. Airy remarks of Whewell's papers that they appear to be "the best specimens of reduction of new observations that we have ever seen."

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1. R is the smaller solar elliptic tide bearing the same relation to T that L does to N amongst the lunar tides. H was omitted as unimportant in schedule [B,i.] of § 23.
2. The tide has been referred by Lubbock and others to an earlier transit, and not to the one immediately preceding the time under consideration. In his case we cannot admit with great accuracy that da/dt = σ, since the interval may be 30 or 40 hours.
3. See Hatt, Phénomène des Marées, p. 151, Paris, 1885.
4. See Prefaces to Indian Official Tide-Tables for 1887.
5. Founded on Whewell's article "Tides", in Admiralty Sc. Manual(ed. 1841), and on Airy's "Tides and Waves," in Ency. Metrop.
6. Waves with a period of from five to twenty minutes are very common, and appear to be analogous to the "seiches" of Geneva and other lakes. See Forel, Bulletin Soc. Vaud. Sci. Nat., 1873, 1875, 1877, and 1879; Ann. Chimie et Physique, vol. ix., 1876; Comptes Rendus, 1879; Arch. Sci., Ph., et Nat., Geneva., 1885; also Airy, "On the Tides of Malta," Phil. Trans., 1878, part i.
7. For a numerical treatment, see Directions for Reducing Tidal Observations, by Commander Burdwood, R.N., London, 1876.
8. An example of this kind of curve for the high-water heights for Bombay drawn automatically by a tide-gauge, would be shown by joining all the high waters together (as in fig. 3) by a continuous curve; and a similar curve may be constructed for the low waters. In this case, however, the hours of the clock are repeated twice over, so that the morning and evening tides occur in different halves of the figure, and the hours are not hours of the moon's transit, but the actual times of high water. It is obvious that the separation of the morning and evening tides prevents the occurrence of the zigzags referred to.