Page:Encyclopædia Britannica, Ninth Edition, v. 23.djvu/385

Rh Schedule of Lunar Tides. [A, i.] Universal Coefficients - T&amp;gt;(- ) Semi-diurnal Tides; General Coefficient = cos 2. Coefficient. "of Argument Speed in Descrip tive Name ~ Degrees per m.s. S Principal K 1-4!2)cO S 4i/ 4542o -2(st } 28—9841042 lunar. Luni-solar (lunar K—&amp;gt; K i+i e2)|sin2/ 03929 30 OS21372 portion). Larger N i ^ cos 4 JI 08796 -2(s-)(s-p) 28 -4397296 elliptic. Smaller elliptic. 1 L I {1-12,5 cos(2p-2^)l i 01257 J where . ^ 29-5284788 Elliptic, 2 I cot2J/-6cos2(j)-^) second JN j A^ pos*ir o L173 -2(s-f&amp;gt;2(s-p] 27 -8953548 order. Larger evectional.2 V ft* -COS^^7 01234: 01.706 -2(!-f)+(j-j o)+2ft -23 28 5125830 Smaller evectional.
 * " &amp;gt; 5 Hour. O

i 13" COS^^f 00176S 00330 -2(s-)-(-p) -2A+ 2S+T 29 -4556254 Variational. 4

i r &amp;gt;*? 00736^ 01094 -2(s-f)+ tt27 -9682084 o, [A, ii.] Diurnal Tides; General Coefficient = sin 2X. Q, Initial. -I Argument t+(h-v). Speed in y. Descrip tive Nairn-. Coefficient. III Degrees per m.s. ari Hour. Q to Lunar di -2(s-|)-(s-p) -(s-^)+Q-iT where tanQ = Jtan(p-0 -2(s-^)+(s-r^) +2/i-2s+37r we urnal. O (l-Je2) s j n / co tyl 18S56 13 9430351 7+20". oo (l-J2)Jsin/sii 12JJ 00812 16-139101&amp;gt; Luni-solar (lunar por KI (l+?e 2 )|sin/cos/ 18115 15-041068( tion). Larger elliptic. Q ^.JsinJcos 2 I 03651 13 398660f me cor ser Smaller elliptic. 9, e.JsinJcos 2 J.Tx 005226 01649 14 492052: mo the y+a-rs. J fe.JsinjfcosJ 01485 15 -585443: no Elliptic, IIP second 7-40-+2CT Ye 2 -isinfcos!JI 00487 12 -854286: uc fll, order. till Evectional. 7-3&amp;lt;r-CT-r277 }?me. J sin/cos 2 } 7 005127 00708 13-471514. eqi Al 3&amp;gt;n [A, iii.] Long Period Tides; General Coefficient rsin 2 X. as su Descrip tive Name. Initial. Coefficient. J 2. ^ Argument. Speed in De grees per m.s. an in of &amp;gt; o 3our. th( O be Change of mean level. (1 + ie2)Kl-?sin 2 /) Of variable 252248 part is N, the long, of node 19 34 per annum to Monthly. Mm 3e.J(l-?sin27) 04136 s 5443747 Th Evectional monthly. &amp;lt;r-217+CJ

M-Kl-fsiua/) 005809 ( -(s-p) 00755 +2s-2h 4715211 ne&amp;lt; to Luni-solar fort MSf 3m2j(i-|sin2r) 004229 2(3-70 1 0158958 fur nightly. 10 00621 an Fort _J? nightly. Mf ( l-fe2)isin 2 / 07827 2(*-&amp;lt;) 1 0980330 Ot th( Ter3 T-CT Je.Jsin 2 / 01516 (-, o)+2(s-s] 1 6424077 mensual. is vai an [B.] Schedule of Solar Tides. Solar Tides; Universal Coefficient =— 2 mc Descriptive Name. Initial. Coefficient. Value of Coefficient. Argu ment. Speed in Degrees per m.s. Hour. [L] Semi-diurnal Tides; General Coefficient = cos 2 X. Principal solar. 82 T i(l _ |e,2) cos* J w 21137 2 30 -0000000 Luni - solar T I (solar por K 2 01823 2t+2/i 30 0821372 tion). Larger el liptic. T T, 01243 2t-(h-p,) 29-9589314 pi.] Diurnal Tides; General Coefficient = sin 2X. Solar diur nal. P ^" (1 - i, 2 )i sin w cos 2 Jw 08775 t-K+fr 14 9589314 Luni-solar (solar por KI - (1 -r-f*, 2 )! sin w cos w 08407 t+h-lir 15 0410686 tion). [iii.] Long Period Tides; General Coefficient =^-f sin 2 X. Semi-an nual. Ssa (1 |e,2)j sin 2 o&amp;gt; 03643 Zh 0821372 1 Fused with 2y-ff+a. 2 m is the ratio of the moon's mean motion to the sun's. 3 In these three entries the lower number gives the value when the co efficients of the evection and variation have their full values as derived from lunar theory. 4 Indicated by 2MS as a compound tide (see below, 24). 5 A fusion of -y -trier, of which the latter is the tide named. 8 The upper number is the mean value of the coefficient of the tide y-ff-rs; the lower applies to the tide MI, compounded from the tides y-ff- CT and 7 The lower number gives the value when the coefficients in the evection have their full value as derived from lunar theory. 8 The mean value of this coefficient is J(l+Se2)(i_3 s i n 2i)(i_^ s in2w)= 25, and the variable part is approximately -(1+ije-) sini cosi sinw cosw cosW= -0328 cos AT. 9 The lower of these two numbers gives the value when the coefficients in the evection and variation have their full values as derived from lunar theory. W Indicated by MSf as a compound tide. From the fourth columns we see that the coefficients in de- Scale of scending order of magnitude are M 2, Kj (both combined), S 2, import0, KI (lunar), N, P, Kj (solar), K 2 (both combined), K 2 (lunar), Mf, ance of Mm, K 2 (solar), Ssa, v, M 1} J, L, T, 2N, /*, 00, 3&amp;lt;7 - nr, tides. 7 - So- - CT + 2i), 7 - 4&amp;lt;r + 2cr, a - It] + CT, 2(ff - 17), X. The tides depending on the fourth power of the moon's parallax arise from the potential Vs-j-p 3 ( cos 3 z - cos 2). They give rise to a small diurnal tide M 1; and to a small ter-diurnal tide M 3; but we shall not give the analytical development.

All tides whose period is an exact multiple or submultiple of a mean solar day, or of a tropical year, are affected by meteorological conditions. Thus all the tides of the principal solar astronomical series S, with speeds 7-17, 2(y-7/), 8(7 -if), &c., are subject to more or less meteorological perturbation. An annual inequality in the diurnal meteorological tide Sj will also give rise to a tide^- - 2tj, and this will be fused with and indistinguishable from the astro P; it will also give rise to a tide with speed 7, which will be indistinguishable from the astronomical part of Kj. Similarly the astronomical tide K 2 may be perturbed by a semi-annual in equality in the semi-diurnal astronomical tide of speed 2(7-77). Although the diurnal elliptic tide Sj 01-7-77 and the semi-annual and annual tides of speeds 277 and 77 are all probably quite insensible as arising from astronomical causes, yet they have been found of sufficient importance to be considered. The annual and semi annual tides are of enormous importance in some rivers, representing in fact the yearly flooding in the rainy season. In the reduction of these tides the arguments of the S series are t, 2t, 3t, &c., and of the annual, semi-annual, ter-annual tides h, 2h, 3h. As far as can be foreseen, the magnitudes of these tides are constant from year year.

We have in § 21 considered the dynamical theory of over-tides. The only tides of this kind in which it has hitherto been thought necessary to represent the change of form in shallow water belong to the principal lunar and principal solar series. Thus, besides the fundamental astronomical tides M 2 and So, the over-tides M 4, Mg, M 8, and S 4, S 6 have been deduced by harmonic analysis. The height of the fundamental tide M 2 varies from year to year, according to the variation in the obliquity of the lunar orbit, and this variability is represented by the coefficient cos 4 /. It is probable that the variability of M 4, M 6, M 8 will be represented by the square, cube, and fourth power of that coefficient, and theory ( 21) indicates that we should make the argument of the over-tide a multiple of the argument of the fundamental, with a constant subtracted.

Compound tides have been also considered dynamically in § 21. By combining the speeds of the important tides, it will be found that there is in many cases a compound tide which has itself a speed identical with that of an astronomical or meteorological tide. We thus find that the tides 0, K 1; Mm, P, M^ Mf, Q, M 1( L are liable to perturbation in shallow water. If either or both the component tides are of lunar origin, the height of the compound tide will change from year to year, and will probably vary proportionally to the product of the coefficients of the component tides. For the purpose of properly reducing the numerical value of the compound tides, we require not merely the speed, but also the argument. The following schedule gives the adopted initials, argument, and speed of the principal compound tides. The coefficients are the products of those of the two tides to be compounded.