GEOMETRICALLY SIMPLE BASINS Narrow Basins.—(r3). In order to elucidate the general characteristics of tides, it is advisable to study the mathematical problems presented by the solution of their determining equations when the conditions are ideally simple. We begin by considering a channel of uniform rectangular section and neglect both the dynamical effects of the earth's rotation and the force of friction. Let x be measured along the length of the channel and y transversely. Then the equations become— for N = i, 2, 3,• • • . When N = i, we have the uninodal seiche; the node is in the middle of the basin, where Kx = 1r, and the amplitude of the eleva tion reaches maxima at the two ends. When N = 2, we have the bi nodal seiche in which nodes occur where KX = 1 71" and Ir. When N = 3, we have the seiche in which nodes occur where KX = z 1r, Ir. In the bi-nodal seiche the motion in each half of the lake is the same as that of a uninodal seiche in a lake of half the length, while in the tri nodal seiche the motion in each third of the lake is the same as that of a uninodal seiche.
Next consider a basin of parabolic longitudinal section such that — xV ho being thus the maximum depth. It is easily verified that the equations are satisfied by— which shows that must be a function of hsecO, or the depth of the ocean measured parallel to the earth's axis. We therefore see that steady currents are possible if there are any closed contour lines of hsecO, and the actual oceans do contain such contour lines. The problem of specify ing the nature of the solution of the dynamical equations for the case of forced long period tides is further complicated by the possibility of reso nance with the free oscillations of the second class. The effects of friction are sufficient to make any forced periodic tide conform to its equilibrium form if its period is sufficiently long, but a criterion of this length has not yet been satisfactorily evolved. If tidal motion were everywhere non turbulent, friction would have no appreciable influence on a constituent whose period was less than a year. It is known, however, that the re sultant motion in shallow seas is turbulent, and it is probable that even in mid-ocean the forces of friction are not less than those calculated on the hypothesis of turbulence.
Phil. Trans. A [1920]). The chief contributory areas were found to be the Bering Sea, the Yellow Sea, Malacca Straitand the North-West Passage, the dissipation in the great ocean basins being supposed in com parison. The problem of estimating the length of time required to destroy the total energy in any constituent, at the rate at which energy is actually being dissipated in that constituent, turns largely on the amount of mag nification of tides in shallow seas. If, in mid-ocean, the semi-diurnal con stituents were of their equilibrium order of magnitude then the time re quired would be only about two days (H. Jeffreys, Nature [1923]).
Compound Constituents.—(r 8). So far in our discussion of tidal dy namics we have neglected terms of the type ax in the equation of continuity, and of the type uou/ ax in the dynamical equations. The ef fect of this has been to make the differential equations linear, and it is the linearity of the equations which allows of the separation into independent harmonic constituents. Such a neglect, however, is not always admissible, especially in shallow water and near rugged coasts. Suppose that over a region there is a predominant harmonic constituent, e.g., M2, of speed n; then the effect of the retention of these terms is that the time enters in the form cos (nt — or 2cos(2nt— E2)+1cos(€1— Terms which contain the time-factor cos(2nt represent a constituent of double the speed of the primary constituent, while those containing the factor cos(E, — represent a steady motion or displacement. The first of these is known as a first overtide and when derived from M2 is called Again, if the principal part of the tides comprises two harmonic constitu ents, then in addition to overtides we have terms of the type— the advancing crest shows that a shorter time elapses from low water to high water than inversely. The investigation also shows that the law of the ebb and flow of currents in estuaries mentioned in §2 must hold good.
Instead of an indefinitely long canal, suppose we take a gulf so that with product terms neglected, we have the solution in which at the head of the gulf = Hcosnt. Then, when we retain the product terms and apply the same process as above, we find at the head of the gulf an additional term of the cos2nt. If II' is greater than IL/ but less than H there will be a double high water (G. B. Airy, Tides and Waves [1842]).