8. Partial Explanation of Ocean From §§ 3, 4 an idea of the magnitude of the equilibrium tide can be obtained. It represents the direct effect of the action of the moon upon the waters where the body is so small and deep that its motion can be ignored. In a larger body whose free period is quite different from that of the tidal forces it is reasonable to sup pose that tides even smaller_ than the equilib rium tides will be raised. In § 7 this was found to be the case for an equatorial canal of mod erate depth encircling the earth. Again, if a canal, bounded at each end, be so shallow that a wave-length ( ) extends through only a few degrees of a great circle, then even if its length approximates to the tides are obviously small because the tidal forces act very nearly alike on both halves, while the particles move in opposite directions in the two halves. Re sults like these contrasted with those obtained from observing the tides of the oceans lead to the belief that, as a rule, the ocean tides as we know them are so great that they can be pro - duced only by successive actions of the tidal forces upon oscillating systems each having, as free period, approximately the period of the forces, and each perfect enough to preserve the general character of its motion during sev eral such periods were the forces to cease their action.
That oscillations according to one of the free periods may persist for a long time can be seen in the case of some seiches; for, although probably started by a meteorological disturb ance and sustained by no periodic force, yet in some harbors, straits, or bays they execute a large number of oscillations before dying out, and their periods are fairly constant in such cases. If a suitable harbor can continue to oscillate for possibly a day or two because of the inertia of the water, it is clear that an oscil lation can be sustained in a large body, like a portion of one of the oceans, by a very small periodic force provided that the free period of the body is nearly equal to the period of the forces and that the boundaries are such that no great amount of energy is carried away by progressive waves or otherwise.
Suppose the oceans by reason of their depths and the configuration of shore lines to contain several, such systems whose free periods are nearly the periods of the tidal forces. These systems generally consist of still more simple sheets having like periods and which may be styled 'areas?) The times of tide can be found by means of the following rule: If to the particles of water in a given oscil lating system, each area of uniform depth, and wherein the resistances are proportional to the velocities of the particles, a series of simple harmonic forces having for period the free period of the body of water be applied, and a permanent state established, then must the time of elongation be simultaneous with the time when the virtual work of the external periodic forces upon the system becomes zero.
The forces in various parts of the system and at various times or hours are those of Fig. 4. projected upon the lines of motion of the par ticles. The virtual displacements are the same at each assumed hour, but differ in different parts of the system.
9. Cotidal Lines.—Although the cotidal lines in the ocean are generally real and distinct, they seldom indicate the progression of a wave at the rate due to depth; such progressions, how ever, exist in many shallow arms of the sea, particularly in tidal rivers. Because the prin cipal ocean tides are due chiefly to stationary oscillations, as has just been pointed out, we may expect to find extensive regions character ized throughout by nearly simultaneous tides. For that portion of the Atlantic Coast of America extending from Rhode Island to the Bahamas and Haiti, high water occurs almost simultaneously. The same is true for south eastern Alaska and the Gulf of Alaska; for the Pacific Coast of Central America, and for the eastern coast of the Philippines. The time of tide changes but little along the Atlantic Coast of Morocco and Portugal. In localities like these the rise and fall amounts to several feet even off shore and in deep water.
The Arctic Ocean is characterized by pro gressive tides of small range derived from the Atlantic Ocean.
In the oceans, in certain arms of the sea, and in certain lakes are to be found a con siderable number of no-tide points due to vari ous causes. From a point of this kind radiate all cotidal lines belonging to a tidal period; and so these lines, or the tide which they repre sent, are there said to be 'xamphidromic.s Excluding lakes and seas not communicating tidally with the ocean, and considering only the semi-daily tides about 25 no-tide points can be enumerated; a large percentage of these are to be found in straits and sounds.
A lake whose longest free period of oscilla tion is several times less than the period of the tidal forces will experience equilibrium tides and possess a no-tide point situated at the centre of gravity of the surface of the lake. The sequence of the cotidal lines about the point will be the same as that of the tidal forces. (Fig. 4). For semi-daily tides the numbers will increase in the clockwise or counter-clockwise sense according as the point lies in north or south latitude.