Tides and Waves. Sect. IV. heory of At low water, the water is not flowing to or from the shore, but is running down the channel. Tides and (363.) Consequently, in the course of one complete tide, the direction of the current will have changed Revolving through 360°, the water never having been stationary. And the direction of the change of current will be of motion of such a kind that, if we suppose ourselves sailing up the mid-channel, the tide-current will turn, in those parts Bubsec. 7. which are on the left hand, in the same direction as the hands of a watch; and in those parts which are on the shore. right hand, in the direction opposite to that of the hands of a watch. o open (364.) Beyond this we can add little to the Theory of Waves upon a sea extended in both dimensions. But the following remarks will be found important with reference to the method of determining from observations some of the phenomena of tides. the tide near the (365.) In tracing the progress of the tide across an extended sea, we cannot observe the different waves as we can those upon a small piece of water. We can do nothing but make observations of the time of the rise and fall of the sea at many different points along the shores of the bounding continents, or at islands in different parts of the sea: and when we have thus ascertained the absolute time of high water at many different points, if they are sufficiently numerous, we may draw lines over the surface of the sea passing through all the points at which high water takes place at the same absolute instant. These lines (adopting the word introduced into Cotidal general use by the highest authority on the discussion of tide-observations) we shall call cotidal lines. The lines on tracing out the cotidal lines in different seas is the greatest advance that has yet been made in the discussion of open seas. the phenomena of the tides in open seas. tion be. tween the (366.) Now when the series of waves is single, the cotidal lines correspond exactly with the lines marking The rela the position of the ridge of the wave at different times. But when the series of waves is compound, it may happen that the form of the cotidal lines will not present to the eye the smallest analogy with the forms of the cotidal ridges of the mingled waves. This will be seen in a simple instance. lines and nate is very (367.) Suppose that there are two systems of parallel waves rolling across the sea at the same time, the the waves ridges of all the waves of one series being parallel to , and the ridges of all the waves of the other being from which parallel to z. The forms of the ridges, as they might be seen by an eye placed at a sufficient distance, would they origi be the intersecting lines represented in figure 19. The elevation of water caused by the former at the time t obscure. will be represented by b.cos nt mz: that caused by the latter will be represented by a.cos nt-pr. We have taken the same coefficient for t in both these expressions, because the recurrence of tide-waves, whether forced waves, or free waves introduced in the integration for satisfying the limiting conditions, must be periodical as the exciting cause. But the coefficients of x and z may be different: thus, for instance, one of these waves may be a forced tide-wave, and the other may be a free tide-wave, in which case the coefficients (292.) will not necessarily be the same: this is, however, immaterial to the present investigation. (368.) The whole elevation, then, of the water at any point will be a sin pr+b sin mz This expression for t determines the time of high water at that place; and, therefore, the line connecting all the points at which it is high water at the same instant will be determined by making equal to the same quantity tan nt. Giving a definite value C to the quantity nt, we have for the equation to a certain cotidal line, = (369.) If a=b, this equation becomes sin mz-C-sin pr-C; whence mz-C-pr+C, or =−pr+C±2%, or-pr+C+4π, &c., or mz-C=pr-C±, or =pr-C±3, &c. These expressions evidently represent two series of straight lines, making equal angles with the co-ordinates: one of them is stationary, (as Ċ or nt disappears from the equation,) and its deduction from the investigation above denotes that there is no sensible tide along those lines: the other is the real system of cotidal lines. The former is represented in figure 20 by the double lines, the latter by the single lines. In drawing the latter, we have supposed C to vary successively byat each step. (370.) If a<b, the first expression for mz-C is included between the arc whose sine is Weber's Wellenrinne. T which values mz Cattains when px-C is ±2 and -±2. The cor T responding curves for values of C varying successively by are represented in figure 21. 2 (371.) If a>b, the curves will be such as those represented in figure 22. (372.) It is evident that the contemplation of these curves (more especially if a small part only of each can be traced) will not easily enable us to discover the nature of the simple interfering systems of waves from which they originate. If either system were complicated, as in (303.), the difficulty would be still greater. SECTION V.-ACCOUNT OF Experiments on Waves. (373.) Our theory of waves, as we have remarked in the Introduction, is not the most complete that could be devised. It embraces (as we believe) every case of general interest to which mathematics are at present applicable, but it does not comprehend those special cases which have been treated at so great length by Poisson (Mémoires de l'Institut) and Cauchy (Savans Etrangers). With respect to these we may express here an opinion, borrowed from other writers, but in which we join, that as regards their physical results these elaborate treatises are entirely uninteresting; although they rank among the leading works of the present century in regard to the improvement of pure mathematics. We shall not therefore trouble ourselves with detailing the few imperfect experiments of Biot (Mémoires de l'Institut) and Bidone (Turin Memoirs) which have been made in verification of these theories. (374.) One of the most important works that has been published, in regard to experiments as well as to the theory of Waves, is that by the two brothers Weber, entitled "Wellenlehre auf Experimente gegründet." This work contains an abstract of all the theories and all the principal experiments of preceding writers that the authors had been able to collect. The points however to which we shall allude here are the experiments made by the Webers themselves. These were made with an apparatus which they call Wellenrinne. It is a very narrow trough with glass sides. In one instance it was 5 ft. 4 in. long, (Paris measure,) about 8 inches deep, and about an inch wide; in another instance it was 6 feet long, 24 feet deep, and a little more than an inch wide. The glass sides were properly supported by pieces of wood connected with the bottom; in the smaller, the glass sides were continuous; in the larger, the glass only occupied 6 openings in different parts of the sides, the other parts being of wood. Some experiments were made with quicksilver, and some with brandy; but the principal part were made with water containing a great number Motion of of floating particles of the same specific gravity as the particles water; by observing the movements of these through floating in the glass sides, sometimes with the naked eye and the water sometimes with a microscope, the motions of the particles of water, even to the bottom, were easily examined. The waves of experiment were generated by plunging a glass tube into the fluid, raising the fluid into the tube by suction, and then allowing it suddenly to drop. observed. wave was obtained; the slate being in this case suddenly plunged into the fluid; these determinations are however confessedly much less satisfactory than the former. The experimenters, however, were able to ascertain that, when the height of the wave was large in proportion to its depth, its front was much steeper than its back, as our theory of (203.) gives it. (376.) The wave was sometimes observed when it had run to one end of the trough and was reflected there, (a method carried to great perfection in Mr. Russell's experiments, to be described hereafter.) But generally the observations were made soon after the wave was formed. We have no doubt that some irregularities in the results were entirely due to the mixture of waves of various lengths which always occurs at first, and that they would have been avoided if the actual observation had been deferred till the principal wave had cleared itself of the small waves. Tides an Waves. Section Experi ments on Waves. Obse (377.) By inspection of the motion of the particles, the Webers discovered the following general rules. When a wave ridge is followed by an equal wavehollow, every particle moves in an ellipse, (or a curve as near to an ellipse as the eye can judge,) whose major axis is horizontal; the motion of the particle lawa when in the highest part of the ellipse being in the moti same direction as the motion of the wave, and in the indis opposite direction when at the lowest part of the parti ellipse. (Fig. 23 is copied from Weber's figure.) When a small wave-hollow follows a large wave-ridge, the motion is such as is represented in fig. 24; and when a large wave-hollow follows a small wave-ridge, the motion is such as is represented in fig. 25. These motions are all in general conformity with the results of our theory in (182.); it being remarked that, by the theory of (226.), &c., the same may (with certain combinations) apply to a single wave. At different They depths the motion was different; the horizontal motion well being diminished in some degree for the deeper par- thed ticles, and the vertical motion being very much diminished, so that, on approaching the bottom, the ellipse became near.y a horizontal line, as shown in figure 26. These results agree with those of (177.), &c. It was also found that different particles in the same vertical line described corresponding parts of their courses at the same instant of time, as we have found in (162.). a suc sion (378.) From contemplation of these experimental Obse circumstances, the Webers constructed figure 27 to moti represent the motion of particles at the surface of a parti progressive wave followed by other waves. We need scarcely point out to the reader that these motions ware coincide exactly with those which we have found in (182.). Irreg (379.) Some discordances were found in the results, rities pendi depending on the manner in which the wave was pro- the w duced, and which it would be extremely difficult to produ the w Jent крегі rats on des and compare with theory. Thus when the suction-tube Wres. was plunged deep in the fluid, it was sometimes found that the horizontal motion of particles near the bottom ion V. was greater than that of particles at about half the depth. The form of the waves was varied by plunging the suction-tube to different depths. When it was very deep, the wave produced was long and flat; when it only touched the surface, the wave was short and high. In the latter case it was found that each particle performed its elliptical revolution in a shorter time than in the former; as the theory of (169.) gives. (380.) In some experiments it was found that the time occupied by particles near the bottom in describing their elliptic courses was less than that occupied by particles near the surface. It is plain that some complicated system of waves was here produced by some peculiarity in the primary disturbance, of which we can give no further account. red (381.) Each particle described its second course in a shorter time than the first. This is evidently caused by a small wave following a large one. (382.) In regard to the general velocity of the wave, the Webers found that it was increased by increasing the depth of the fluid in the trough, but they did not ascertain the law. They also found that it was independent of the specific gravity of the fluid. They found that a bulky wave travels more quickly than a small one, as appears from (208.). (383.) Observations were also made of the motion of of the particles when two equal waves meet each other. It was found here that the motion of each particle was MSWO backwards and forwards in a straight line, as is represented in fig. 28, which is copied from Weber's figure. We need scarcely to point out that this is precisely the same kind of motion as that which we have found from theory in (189.), &c. (384.) Other observations were made by the Webers, but none which seem to bear closely upon our theory. (385.) In regard to the experiments that we have abstracted, we may give our opinion as follows:-The contrivance of using a vessel with glass sides and observing the motions of floating particles is one so admirably adapted to overcome the greatest of all the difficulties attending the comparison of a wave-theory with experiment, namely, that of ascertaining the laws of movement of individual particles, that we think it gives these experiments a claim for superiority above all others. In other respects we think causes of uncertainty may be pointed out. The narrowness of the troughs used makes the effect of any irregularity of the sides great. The rapidity of the observation throws great doubt on the measures of time. How ever much the Webers might be inclined to trust to their "Tertien-Uhr," (a watch with which the part of a second of time could be observed,) we have little confidence in the use of it. The same cause-namely, the observation of the waves as soon as they were formed-has introduced great complexity into the facts of experiment, which would not have existed if the slower process used by Mr. Russell had been adopted. Although a complete theory ought to explain the most complicated experiments, yet, under all the difficulties of wave-mathematics, we must confine ourselves to simple cases if we wish to have valid comparisons of theory and observation. We would however point out to any future observer the use of VOL. V. Waves. Weber's Wellenrinne with several of Russell's methods Tides and of observation as likely to give better results than any yet obtained. (386.) No allusion is made to theory, in the course of the Webers' experiments; and though they have stated the leading points of several theories, (in another part of their book,) they do not appear to have the power of familiarly applying them. We look upon their experiments therefore as quite free from theory, and for that reason we consider their coincidences with theory as peculiarly valuable. ments. (387.) Mr. Russell's experiments on Waves are con- Russell's tained in the Report of the Seventh Meeting of the experiBritish Association, p. 417-496. They constitute, upon the whole, the most important body of experimental information in regard to the motion of Waves which we possess. We shall endeavour here to epitomize the principal contents of that paper, (omitting, for the present, all that relates to the tide-wave;) it will be necessary, however, to make some remarks upon Mr. Russell's references to theory, because we believe that any one who should derive his first knowledge of the nature of waves from that paper would receive from it a most erroneous notion of the extent of the Theory of Waves at the date of those experiments. wave. (388.) We shall commence with the experiments Apparatus made with apparatus arranged expressly for this pur- for crepose. A rectangular trough or cistern was constructed, ating a 20 feet long, 1 foot broad, and more than 7 inches deep. At one end, an additional length of 7.3 inches was left, so that in fact the trough really was an uninterrupted trough, whose length was 20 feet 7.3 inches. Only 20 feet, however, was used in the experiments, the remaining part being used for the generation of a wave, in one of the following manners. A sluice being placed at the distance 7.3 inches from the end, water was poured into the small part of the trough behind the sluice, to a known height above the surface of the water in the trough; then, upon raising the sluice, that portion of this water which was higher than the general level (and whose volume therefore was known) rushed into the trough, forming a swell there which was immediately propagated as a wave along the surface of the water in the trough; and the sluice, being depressed, formed a smooth end to the trough in that part from which the wave began. Or, a vertical rectangular trunk, occupying the whole or a part of the small portion at the end of the horizontal trough was filled with water to a certain height, and, by lifting the trunk, that water was allowed to gush out below its lower edge. Or, the sluice of which we have spoken was used to form a wave by merely agitating it with the hand. And in some experiments the disturbance was given by pressing a solid into the water, and in others by withdrawing a solid from the water. of a wave. (389.) The method used for measuring the velocities Mode of of the waves is extremely ingenious. The length of increasing 20 feet was far too small to permit of any accurate de- the range termination of velocity. But Mr. Russell remarked that the wave, upon meeting one of the vertical ends of the trough, was reflected without alteration of form, and therefore could be observed in its reflected course as well as if the trough had been prolonged; and, as the same remark applied to every reflection at each end of the trough, the trough might be used as a channel of indefinite length. (The theory of (355.) and (357.) shows that the reflection from the plane end will in all cases 2 z* Waves. Tides and produce a wave of exactly the same kind as that which comes in contact with the end, whatever that kind may be.) Thus the wave was sometimes observed after it had been reflected 60 times, or after it had really described a length of 1200 feet. Moreover, the progress of the wave was observed without difficulty at a great number of points in its course, for instance, in the experiment just cited, at three points in each length of 20 feet, or in 180 points in the length of 1200 feet. The first observations were usually made after the wave had run the length of the trough once or twice; this allowed many small waves (such as apparently have injured Weber's experiments) to separate themselves and disappear. Mode of observing the passage of a wave. Section ments of Wares of the length the wa upon in (234.) shows that such a wave may travel, without any Tides λ k (390.) The method of observing the time at which the crest of a wave passed a given point was most happy. The flame of a candle, placed above the trough and at a small horizontal distance from it, was reflected by a mirror in an inclined position downwards to the water, then by the surface of the water it was reflected upwards, and being received upon another inclined mirror was reflected to the eye of an observer, who viewed it through an eye-tube, furnished with an internal wire and a more distant mark for directing the observer's eye. When the water was at rest, or when the horizontal surface at the top of the wave was passing under the mirror, the candle was seen in the centre of the eyetube; when an inclined part of the wave (either the anterior or the posterior) was passing, the candle was seen on one or other side of the eye-tube. In this manner the passage of the highest part of a wave whose length was three feet, and whose height was only onetenth of an inch, could be observed with (391.) The length of the wave was observed by observing adjusting two fine conical points, which nearly touched the length, the quiescent surface, so that the anterior part of the of a wave. wave would touch one and the posterior part would leave the other at the same instant. The height of the wave was observed by noting the elevation of the water in small pipes passing from the side of the trough and turning upwards at its outside. We doubt the accuracy of these determinations; they are, however, less important than the determination of velocity; yet we shall presently find that fuller information regarding of 3bk we must put them would have been valuable. Mode of and height Species of wave ob Russell. accuracy. (392.) Mr. Russell's researches, in these experiments, were directed entirely to the examination of what he deserved by nominates "The great primary wave," and which he describes as “" differing in its origin, its phænomena, and its laws, from the undulatory and oscillatory waves which alone had been investigated previous to the researches of Mr. Russell." We are not disposed to recognize this wave as deserving the epithets "great" or "primary," (the wave being the solitary wave whose theory is discussed in (226.) &c.,) and we conceive that, ever since it was known that the theory of shallow waves of great d2X d2X length was contained in the equation =gk (195.), di2 dx with limitations similar to those in (226.), the theory of the solitary wave has been perfectly well known. Leaving this, however, we may state that Mr. Russell's experiments were all made upon a single wave of considerable length, similar to that discussed in (230.) and (232.), in which a particle is actually moved a certain distance by the wave and then remains at rest in a position differing from its original position. The result the m (393.) There is, however, another point to be con- Infu sidered, namely, that the height of the wave, in many of the of the experiments, bears a sensible proportion to the heigh depth. According to the theory of (208.), supposing upon the succession of waves continuous, the top of the wave velo would travel with a velocity greater than that due to the undisturbed depth, and even greater than that due to the disturbed depth, and expressed by gk × (1+3b), height of wave where b But if, as in continuous depth of water' waves, we refer our first calculation not to the undisturbed depth but to the mean depth; then instead of k we must put k 1 + the mean depth; and instead 2 2 Thus the last formula becomes b 36 gk1 + (394.) To examine, then, the general coincidence of Mr. Russell's results with the theory, we have proceeded thus:-We have taken the abstract in pp. 440, 441, 442, of the Report of the British Association, having corrected a few errors in it, and have divided the experiments into groups in which the depth of the water and the height of the wave are nearly equal. We have assumed that the mean of the observed velocities corresponds to the mean of the depths, &c., an assumption which is not rigorously true, but probably much nearer to truth than any one experiment. We have then computed the theoretical velocity for the undisturbed depth by the formula of (169.), &c., supposing λ=36 inches; and in other columns we have altered this velocity in the proportion of 1: √1+6, 1: √√1+35, and 1: √1+26. relo 6.69 to 7 20 7.74 to 8.00 (395.) The experiments which are most favourable agree for determining the influence of the height of the wave aly are those of the second group. If we compare the those column of "Velocity computed for undisturbed depth" uted with the column "Observed velocities," we find that all the computed velocities are too small. If we compare the "Computed velocity × √1+b," which is the same as that due to the depth measured from the crest of the wave to the bottom of the trough, we find that 9 are too small and 3 too great. If we compare the Computed velocity x 1+26," which is that deduced from our theory of (208.), we find that 5 are too small and 7 too great. If we compare the "Computed velocity × √1+3b," we find that 3 are too small and 9 too great. The comparison of the first group leads to nearly the same result; the numbers in the corresponding columns being all too small - 6 too small, 2 too great 4 too small, 4 too great 4 too small, 4 too great. On the whole, therefore, we think ourselves fully entitled to conclude from these experiments that the theory of (208.) is entirely supported; and that the velocity is correctly calculated by supposing it to be that due to the mean depth increased by three times the semi-oscillation in depth, or the whole depth from the crest of the wave increased by the whole oscillation in depth. (396.) The reader will, however, remark that the ming excesses of our computed quantities are for the most re-part in the small depths of water, and the defects in the by great depths. We think it most likely that this is due leto the difference in the lengths of the waves. es. It is not unlikely that A was less than 36 inches in the small depths, and greater than 36 inches in the great depths. 3.586 3.808 4.216 4.017 If we had calculated with such numbers, we should the length have found smaller computed velocities for the small of the depths, and greater for the great depths; and the waves. agreement with the observed velocities would have been extremely close. character. (397.) Other experiments of Mr. Russell's were Experidirected to the inquiry, whether the mode of producing ments of the wave (in other words, the form of the wave) in a general fluenced its velocity; it was found that no difference of velocity was perceptible with waves produced in different ways. This is in accordance with (234.). Experiments were also made, (of which no details are given,) which showed that the motion of the particles from the surface to the bottom of the channel is the same, and that particles once in a vertical plane continue in a vertical plane. These results agree with those of (180.). (398.) Some experiments were made by Mr. Russell on what he calls a negative wave-that is, a wave which is in reality a progressive hollow or depression. But (we know not why) he appears not to have been satisfied with these experiments, and has omitted them in his abstract. All the theories of our IVth Section, without exception, apply to these as well as to positive waves, the sign of the coefficient only being changed. We may remark, as a matter which may be observed Negative (in some localities) in daily experience, that the phæno- wave promenon of a negative wave is given in great perfection duced by by the paddles of a steam-boat: the first wave which of a steamthe paddles passes away from it being a hollow of considerable boat. depth. We were first made aware of this by observation of the traces made by Mr. Bunt's excellent selfregistering tide-gauge on the banks of the Avon, at a short distance below Bristol; but we have since fre |