PHILOSOPHICAL TRANSACTIONS.

I. THE BAKERIAN LECTURE.-Researches on the Tides. Thirteenth Series.

On the Tides of the Pacific, and on the Diurnal Inequality.

By the Rev. W. WHEWELL, D.D., F.R.S., &c.

Received November 11,-Read December 16, 1847.

1. IN 1833 the Royal Society did me the honour to publish, in its Transactions, a memoir of mine, entitled "Essay towards a First Approximation to a Map of Cotidal Lines;" and, in subsequent years, a number of further communications on the subject of our knowledge of the tides, as deduced from observations of those phenomena. These later "Researches" have modified my first views,-a result which I from the first contemplated as probable, as I intended to imply by entitling my memoir "An Essay towards a First Approximation," and as I expressed more fully in the memoir itself. I have also obtained from various persons, since my last communication to the Society, a considerable amount of recent tide observations, made in various quarters. of the globe; and I am desirous of pointing out the general bearing of these additional materials of knowledge. I wish especially to bring under the consideration both of mathematicians and of navigators, the problem of the tides of the Pacific Ocean. When I wrote my first memoir on the subject, our knowledge of the tides of that ocean was so imperfect, that I did not even venture upon a first approximation to the cotidal lines. And I have since seen reason to believe that, not only for that ocean but for all large seas, the method of drawing cotidal lines which I formerly adopted, is very precarious.

2. There is another leading feature of the tides, which has been brought clearly into view in the course of these researches, which is of great interest and importance to the navigator, as well as to the mathematician, and of which I have assigned the laws in a general manner, and with an accuracy sufficient for most practical purposes; I mean the Diurnal Inequality which makes the common or semidiurnal tides differ alternately in excess and in defect. I have already examined various series of

tide observations in which this diurnal inequality prominently appears; but I have now the means of showing it to be much more extensively distributed and larger in amount than has been supposed.

These two points, the Cotidal Lines, and the Diurnal Inequality, will be the subject of the present memoir.

Of Cotidal Lines.

3. Great light was thrown upon the form of the cotidal lines by very extensive series of observations made for that purpose in June 1834 and June 1835, by the Preventive Service at all their stations on the coasts of England, Ireland and Scotland, and by naval officers at many points of North America, Spain, Portugal, France, Belgium, the Netherlands, Denmark and Norway. These observations, made at my suggestion, by the kindness of the authorities of that time, were so numerous and exact as to determine with considerable accuracy the form of the cotidal lines in the neighbourhood of the oceanic coast of Europe. One main feature, very prominent in all these lines, was that they meet the shore at a very acute angle, and follow its flexures at a little distance with an almost parallel course; and that consequently, the tide-wave which runs up a channel is, in the middle of the channel, very much in advance of its place at the sides of the channel *. This form of the cotidal lines is also easily shown to be in harmony with the laws of the motion of fluids; and it cannot be doubted that those lines must affect such a form to a much greater extent than was assigned to them in my First Approximation.

This character of the cotidal lines must prevail to such an extent that I conceive all attempts to draw such lines across a wide ocean by means of observations on its shores, must be altogether worthless. This applies beyond doubt to the Pacific Ocean, and probably, taking other reasons into account, to the Atlantic also.

4. This conclusion is further confirmed by our finding that if we do draw "cotidal lines" across wide oceans, as for instance, the Atlantic, they do not agree with tides observed at islands in the mid-ocean, without ascribing to the lines such flexures as deprive them of all simplicity, and make them require further evidence.

5. Again, it is found that, for the most part, the tides in the mid-ocean isles are very small; and this circumstance again, makes the assumed oceanic continuity of cotidal lines very doubtful.

6. Further if the tides in the Atlantic and Pacific be conceived to be brought by a progressive wave, which the scheme of cotidal lines assumes, they must be conceived to be brought from some part of the ocean where such a wave can travel round the globe of the earth so as to follow the moon, or at least, to be connected with such a part of the ocean: and such a supposition was accordingly involved in

* See the Charts of the British Isles and of the Coasts of Europe in the Sixth Series of Tide Researches, Philosophical Transactions, 1836, Part II.

† See Mr. AIRY on Tides and Waves, Art. 359, in the Encyclopædia Metropolitana.

the attempts to draw the cotidal lines of the Atlantic and Pacific. But it appears unlikely that this supposition rightly represents the mode in which the waters of the Atlantic and Pacific obey the action of the sun and moon.

7. If it be asked what other mode of operation of the lunar and solar forces upon the ocean can be conceived, different from this progressive wave which is expressed by means of cotidal lines; an answer immediately suggests itself, that a stationary undulation, corresponding in its period with the period of the moon's apparent revolution, from meridian to meridian (that is, a lunar half-day), is a possible mode of motion for a fluid under such circumstances. By a "stationary undulation,” I mean a motion such as that which takes place in a vessel of water, when one side is suddenly lifted from rest, and then set down again. When this is done, the water oscillates, the surface rising alternately on the raised side and on the other, and the middle line of the surface neither rises nor falls. In this case the oscillation is free, depending on the dimensions of the fluid only: but if the fluid were subject to periodical forces, such as an attracting body passing over it at equal intervals of time, it might perform forced oscillations of the same kind; and in this case, the period of the oscillation would necessarily, in the ultimate condition of the fluid, be the same as the period of the forces.

Fig. 1.

8. The lunar attraction passes over every wide ocean once in every lunar half-day; and it is conceivable that such an ocean, under the influence of the lunar forces, should perform, every lunar half-day, such a stationary oscillation as has been described. On this supposition, we should have a regular tide at its eastern and western shore, but no tide in the middle part; and in such a case there would be no cotidal lines. The ocean would be divided into two portions or areas by a line of no tide (AB); and these two areas would each have the tide over the whole area at the same time, the two times differing by six lunar hours. For instance, if the time of high water on the eastern shore, ACB, were one o'clock, the time of high water on the western shore, ADB, would be seven o'clock. This might be expressed by distinguishing the two spaces by shading, and marking them I. and VII. respectively. (See fig. 1.)

II

C

VII

B

Fig. 2.

A

D

VIL

VII

VIL

E

B

9. But this cannot be a representation of the state of the tides of oceans generally, at least as to their littoral spaces. For we know by observation that, along large tracts of the shores of all seas, the tide does travel progressively, in such a manner that its course in those parts may be represented by a series of cotidal lines. And if the shore be broken by shallow inlets and bays, the tide must, by the laws of fluids, travel progressively up these recesses. cases, we may properly represent the state of the tides by such a diagram as before, bordered with a series of cotidal lines representing the course of the tide into the

In such

recesses of the coast; which will be expressed by drawing the cotidal lines of two hours, three hours, &c. (II. III., &c.) belonging to the tides derived from the oceanic tide I.; and by drawing the cotidal lines of VIII. IX., &c., derived in like manner from the oceanic tide VII. (See fig. 2.)

10. But, moreover, if CD be the shore transverse to the line of no tide AB, it is very possible that the tide may not be absolutely simultaneous from C to D, but may vary continuously along this shore, although the tide in the central oceanic parts be of the nature of a stationary undulation, as already supposed. In this case, the state of the tides will require that, in Fig. 3. a map, we should place the extremity A of the line of no tide at some distance from the shore, and there will be a series of cotidal lines I. II. III. IV. V. VI. VII., which will revolve about the point A, and will carry the tide in succession to all the points of the coast CD. These lines will have the general form of cotidal lines which we have described; and will be determined in their details by the shore along which the progressive tide-wave travels. (See fig. 3.)

D

I

VII

B

VII

11. But there is yet another modification which the cotidal spaces of the oceanic tides undergo, in passing into the cotidal lines of littoral tides. The oceanic tides produced by a stationary undulation, with its midline of no tide, give tides necessarily differing six lunar hours from each other on the opposite sides of the ocean. It is high water on one side when it is low water on the other. Now this cannot be the case all over an ocean which is of different breadths in different parts; nor, in fact, can it be the case in any part of it. For that motion of the parts of the fluid which a stationary undulation requires, cannot take place on a shore shallow in proportion to the depth of the oceanic spaces. Near the shore, we shall have a tide which is progressive from the oceanic space towards the land. Hence, even when the tide occurs at the same time along a great extent of shore on one side of an ocean, we cannot assume that it is directly produced by the oceanic tide. It may be a tide which is later than the oceanic tide, and which may be represented by a cotidal line bordering the space which the oceanic tide occupies. (See fig. 4.)

12. And this may be the case with regard to a detached island, as well as to an extensive coast; especially if the island be the summit of an extensive part where the ocean grows shallower. In such a case the island may have about it cotidal lines in the form of rings. (See fig. 4.)

13. From these considerations, it appears that it must be very difficult to determine the time of high water in the oceanic spaces. All that can be certainly known is, that the time must be earlier than the earliest neighbouring littoral tides. And such an oceanic tide being assumed, the

IV

Fig. 4.

sequence of the tide-hours along the shore may be rightly represented by cotidal lines drawn nearly parallel to the shore, according to the data which observation supplies.

14. It is to be observed, however, that the oceanic tide necessarily includes two different areas, in which the times of high water differ by six lunar hours; and therefore that which is assumed as the oceanic space, must be so situated that the tides on its opposite sides differ by about six lunar hours. The oceanic space thus occupied by the stationary undulation must, further, be so situated that the tides in its middle parts are small or disappear.

Fig. 5,

15. But a stationary undulation, such as has been supposed to occupy the oceanic space, is not the only mode in which we may conceive large littoral tides combined with small central tides. Such a combination may be produced by cotidal lines revolving round a fixed centre, as in the upper part of figure 3. Nor is such a state of things imaginary only. The course of the tides on the opposite coasts of England and the Netherlands is such as not to be intelligible in any other manner than by supposing such a rotation of the tide-wave, as may be seen in my Sixth Series of Tide Researches. And the smallness of the tide in the central parts, which this view of the subject implies, has been verified by the observations of Captain HEWETT and others. A case somewhat of the same kind occurs on the coast of Ireland. At Courtown on the east coast, there is no lunar tide*, though there is a progressive tide-wave on the opposite coast of Wales.

VIII

VI

IX

IV

16. It may be thought, therefore, that we shall find it impossible to decide whether the tides of an ocean in which the central tides are small and the littoral tides progressive, are to be represented by a revolving wave or by a stationary oceanic undulation with bordering cotidal lines. But this difficulty is not of much real consequence; for one of these hypotheses passes gradually into the other, as may be seen in fig. 3. And the result of both the one and the other is, that we cannot pronounce anything certain about the time of high water in the oceanic space, till we have been enabled, by numerous observations, to draw the littoral cotidal lines with considerable accuracy; and when this is done, the nature of the oceanic movement will probably show itself upon the face of our chart.

17. On these grounds, I am now disposed to retract parts of what I have said with regard to the form of the cotidal lines of the Atlantic in my "Essay." I do not think it likely that the course of the tide can be rightly represented as a wave travelling from south to north between Africa and America. We may much better conceive the state of things by means of a stationary undulation, of which the middle space is between Brazil and Guinea, in which region the tides are very small, as at St. Helena * See Mr. AIRY in the Philosophical Transactions, 1845, Part I.