ON TIDES. AND TIDAL ACTION IN HARBORS.*

By PROFESSOR J. E. HILGARD, OF THE U. S. COAST SURVEY, WASHINGTON, D. C.

LADIES AND GENTLEMEN: I propose to engage your attention this evening with the subject of the tides of the ocean and the influence exerted by tidal currents on our harbors. I shall first briefly describe the phenomena of the tides as they present themselves to an observer, then consider the physical causes to which these phenomena are due, next examine more in detail the phases of the tide on our own coasts, and finally describe the tidal hydraulics of the magnificent harbor of New York.

The most obvious change in the surface of the ocean to be noticed upon our shores is the alternate rising and falling regularly twice in every day. Closer attention will show that the tides of each day occur somewhat later than those of the preceding day, the average time of retardation being fifty-two minutes, and that this retardation corresponds to that of the moon. It will pass as a fair approximation to say, that it is high water at New York with a southeast moon, or similarly for New Castle, on the Delaware, that high water occurs when the moon is south In fact, so closely is the time of tide connected with the position of the moon, that in order to give the time of high water upon any day approximately it is customary to state the time of high water on the days of the new and full moon, when the moon passes the meridian at twelve o'clock, nearly. This time is called the "establishment of the port." Then, to find the time of high water on any other day, it is only necessary to add the "establishment" to the time of the moon's meridian passage on that day. On closer examination, it will be found that the interval between the time of the moon's passage over the meridian and the time of high water, called the luni-tidal interval, varies with the moon's age very sensibly. Moreover, the elevation at high water and depression at low water will not always be the same, but will be greatest about the times of new moon and full moon, and least about the first and third quarters. The details of these variations will be best traced out in connection with the explanation of their causes, to which we will now proceed.

The popular explanation of the tides, as depending on the law of gravi tation, is sufficiently simple, although the complete mathematical investigation of the subject, by which we should be enabled to predict their * Delivered before the American Institute, January 27, 1871, with revision.

occurrence and magnitude for any place, is encompassed with difficul ties, from causes to which we shall hereafter revert.

If we conceive the earth to be wholly, or for the greater part, covered with water and subject to the attraction of the sun, the force of which varies inversely as the square of the distance, it will be obvious, that while the whole earth will fall toward the sun with a velocity proportioned to the aggregate attraction upon its solid portions, (which is the same as if all the matter were collected at its center,) the water nearest to the sun being accelerated by a greater force, and being fluid, will approach the sun more rapidly than the solid core. It will thus run from all sides into a protuberance beyond the form of equilibrium of the earth's attrac tion and rotation, until the pressure of the elevated mass equals the dif ference in the attraction of the sun. Moreover, a similar protuberance will be formed on the side opposite to the sun, since the particles of water, being solicited by a less force than the solid core, will fall more slowly toward the sun, and as it were remain behind. Nor does the fact that on the average the earth does not lessen its distance from the sun, in the least invalidate the force of this reasoning; for the deviations from the tangential motion of the earth in its orbit are precisely those which the earth would move through if falling toward the sun unaffected by any other impulse.

The same considerations hold good in regard to the attraction of the moon upon the earth and the waters surrounding it; for although we are in the habit of considering the moon as simply revolving about the earth, it must be remembered that the attraction is mutual, that both bodies describe orbits about their common center of gravity, and that while the moon obeys the attractive force of the earth, the latter equally follows that of the former, by which it is at every instant of time drawn from the path which it would pursue if that influence did not exist by an amount precisely equal to the fall corresponding to the moon's attractive force.

As a necessary consequence of the elevation of the water in the regions nearest to and most remote from the attracting body, there must be a corresponding depression below the mean level of the sea at points distant ninety degrees from the vertices of the protuberances, or at the sides of the earth, as seen from the sun or moon. If the latter bodies maintained a constant position with respect to the earth, the effect would therefore be to produce a distortion of figure in the oceansurface, (assumed to cover the whole earth,) having the form of a slightly elongated ellipsoid, the two vertices of which would be the one precisely under, the other precisely opposite to, the points at which the disturbing body is vertical. This, however, is not the case; for by the rotation of the earth, and the motion of earth and moon in their orbits, the direction of the disturbing forces is constantly changing with respect to any point on the earth's surface. New points arrive at every instant under the zenith and nadir of either luminary, and thus it is that waves are produced which follow them round the globe. The highest points

of these waves will remain far behind the verticals of the disturbing bodies, because the inertia and friction of the water prevent the rapid change of form required, and because, although the elevating force is greatest under the vertical, it still continues to act in the same direction for some hours after the passage of the luminary, with but little diminished force.

This retardation, which would be sensible under the simple supposition of an uninterrupted ocean covering the earth's surface, becomes very considerable under the actual circumstances of the case. The depth of the sea varies so much, and the form of its basin, taken as a whole, is so interrupted by the land, that no regular progressive movement of the tide-wave can take place, except in the great Southern Ocean. At all points on the coast the phases of the tide will follow the periodicity of the forces causing them, but at each point, at a greater or less interval from the culmination of the sun or moon, according to its local position, and the more or less circuitous course taken by the tide-wave to reach it. This interval and the actual rise and fall of the tide must be determined for each place by special observation.

LUNI-SOLAR PHASES OF THE TIDES.

The close relations which the times of high water bear to the times of the moon's passage show that the moon's influence in raising the tides must be much greater than the sun's. In fact, while the whole attraction of the sun upon the earth far exceeds that of the moon, yet owing to the greater proximity of the latter, the difference between its attraction at the center of the earth and at the nearest or most remote point of its surface, which difference alone produces the tides, is about two and a half times as great as the difference of the sun's attraction at the same points.

SEMI-MONTHLY INEQUALITY.

We will now consider the particular phases resulting from the combination of the lunar and solar tides, and from the varying positions of those bodies. There will be two complete lunar tides in every lunar day of twenty-four hours fifty-two minutes, and also two complete solar tides in every mean solar day of twenty-four hours. These are known as the semi-diurnal tides, and constitute the principal variations of the sea-level. The combined effect of these two fluctuations will be most readily understood by reference to the annexed diagram, in which the lunar tide is represented by dashes, the solar by dots, and the combined or actual tide by a full line. At the time of syzygies, or full and change of the moon, the effects of both sun and moon combine together to produce the spring-tides, when high water is higher and low water is lower than at mean tides by the amount of the solar tide. At quadratures the high water of the sun will combine with the low water of

the moon to produce a less fall, and the low water of the sun with the high water of the moon to produce a less rise than at mean tides; and we have the neap-tides, the range of which is less than the mean range by the amount of the solar tide. Thus, at New York, the rise and fall at syzygies is 5.4 feet, at quadrature 3.4 feet, the former being the sum,

the latter the difference of the lunar and solar tides, whence we obtain for the effect of the moon 4.4 feet and for that of the sun one foot, or a ratio of forty-four to ten. This proportion does not prove to be the same in all parts of the world, and even varies considerably in places

not far distant from each other. At Boston the heights are 11.3 and 8.5 feet, respectively, giving a proportion of seven to one. On the Atlantic coast of the United States it averages about five to one, while on the east side of the Atlantic Ocean, on the coasts of France and England, it is in many parts as three to one. These differences are to be ascribed to the fact that the shore and harbor tides which we observe bave in every instance acquired a greater magnitude than the ocean tides, in consequence of the wave having passed over a sloping bottom and having been greatly retarded by the effect of friction. A comparison of the range of spring and neap-tides, therefore, will not serve as a correct measure of the relative effect of the sun and moon, unless the effect of friction were taken into consideration, which we are at present unable to do for want of a complete knowledge of the configuration of the bottom.

The interval between the moon's meridian passage and the time of high water is subject to a variation similar to that of the height. On the day after the spring-tides, the top of the solar tide-wave will be nearly an hour in advance of the lunar tide-wave, and the two waves will combine to make high water earlier than the moon's alone would bring it. It will continue to be earlier until the moon's transit is later by three hours, or in the first octant. It then falls back until it is latest in the third octant, and again advances, until, at the next spring-tides, it reaches its mean period. The mean of all the luni-tidal intervals for half a month at a port its called its mean establishment, which is used for finding the time of high water on any given day; and tables are constructed from observations at the principal ports for finding the correction for semi-monthly inequality due to the moon's age. Thus, for New York, the mean luni-tidal interval is Sh. 13m., and its least and greatest values are 7h. 52m. and Sh. 35m. On the Atlantic coast of the United States the range of this inequality is about threequarters of an hour; on the coasts of France and Great Britain it often exceeds one and a half hours.

DIURNAL INEQUALITY.

The next variation of the tides to be considered is that dependent on the moon's declination. Were that body constantly in the plane of the equator, the highest points of the tide-waves would also be in that plane, and would consequently produce a series of equal tides at any place either north or south of the equator. But it is evident that, when the moon ascends to the north, the vertex of the tide-wave will tend to follow it, giving the highest point of one tide in the northern, and the highest point of the opposite tide in the southern, hemisphere. Consequently, when the moon has a northern declination, the tide at any place in the northern hemisphere caused by its upper transit will be higher than that caused by its lower transit. (See diagram of diurnal inequality.) This variation in the heights has a period of one lunar day, and