A STUDY OF SPLASHES

BY THE SAME AUTHOR.

A FIRST COURSE OF PHYSICAL LABORATORY PRACTICE.
Containing 264 Experiments.
With Illustrations. Crown 8vo, 4s. 6d.

DYNAMICS OF ROTATION.
An Elementary Introduction to Rigid Dynamics.
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LONGMANS, GREEN, AND CO.
LONDON, NEW YORK, BOMBAY, AND CALCUTTA.

Permanent Splashes left where a Projectile has entered an Armour-plate.
[See [page 120.]

A STUDY OF SPLASHES

BY
A. M. WORTHINGTON

C.B., M.A., F.R.S.
HEADMASTER AND PROFESSOR OF PHYSICS
AT
THE ROYAL NAVAL ENGINEERING COLLEGE, DEVONPORT

WITH 197 ILLUSTRATIONS FROM INSTANTANEOUS PHOTOGRAPHS

LONGMANS, GREEN, AND CO.
39 PATERNOSTER ROW, LONDON
NEW YORK, BOMBAY, AND CALCUTTA
1908

All rights reserved

DEDICATED
TO
THE NATURAL HISTORY SOCIETY OF RUGBY SCHOOL
AND ITS FORMER PRESIDENT
ARTHUR SIDGWICK
IN REMEMBRANCE OF THE ENCOURAGEMENT
GIVEN TO THE EARLY OBSERVATIONS MADE IN BOYHOOD
BY MY OLD SCHOOL-FRIEND
H. F. NEWALL
FROM WHICH THIS STUDY SPRANG.

PREFACE

This publication is an attempt to present in a form acceptable to the general reader the outcome of an inquiry conducted by the aid of instantaneous photography, which was begun about fourteen years ago. The author, in 1894, had occasion to lecture at the Royal Institution on the "Splash of a Drop," of which he had already made a somewhat prolonged study. That lecture, which was subsequently reprinted in the "Romance of Science" series by the Society for Promoting Christian Knowledge, dealt largely with the splash of a drop falling on a solid plate, with which the present volume is not concerned. At the close of the lecture were exhibited for the first time a few photographs of some of the phenomena now dealt with, which the author had just succeeded in taking with the help of his friend Mr. R. S. Cole. The success of the photographs and the additional information they afforded led to a long photographic investigation, which formed the subject of two papers[A] in the Transactions of the Royal Society. Except for two magazine articles,[B] the results of this work have not been presented to the general public. Moreover, in the illustrations printed by the Royal Society much of the beauty of the original photographs was lost in the reproduction, or was sacrificed in a selection of which the only object was the elucidation of points of technical scientific interest.

If the present volume is so fortunate as to find many readers among the general public, as the author hopes it may, especially among the young whose eyes are still quick to observe, and whose minds are eager, it will be on account of admiration for the exquisite beauty of some of the forms assumed, of surprise at the revelation of so much where so little was expected, and because of the peculiar fascination that is always felt in following any gradually changing natural phenomenon, in which the sequence of events can, partly at any rate, be anticipated and understood.

For the sake of serious students of Physics who may be interested in unexpected phenomena of fluid motion, all references that seem necessary have been given in footnotes, and it may be mentioned that the later photographs of Series I and those of Series Ia and III, have not been previously published, and afford new information on certain points.

In taking these photographs the author has been much helped by his friends Dr. G. B. Bryan and Mr. G. F. Page.

A. M. W.

Tavistock, Sept. 18, 1907.

[Added March 1, 1908.] A slight delay in the publication of this book has afforded the opportunity of obtaining the new and quite unexpected information given in the supplementary chapter.

FOOTNOTES:

[A] "Impact with a Liquid Surface," by Worthington and Cole. Phil. Trans. Roy. Soc., A 193, 1897, and A 255, 1900.

[B] Pearson's Magazine, July and August, 1898.

CONTENTS

A STUDY OF SPLASHES

CHAPTER I

PRELIMINARY—METHODS OF OBSERVATION AND APPARATUS

There will be but few of my readers who have not, in some heavy shower of rain, beguiled the tedium of enforced waiting by watching, perhaps half-unconsciously, the thousand little crystal fountains that start up from the surface of pool or river; noting now and then a surrounding coronet of lesser jets, or here and there a bubble that floats for a moment and then vanishes.

It is to this apparently insignificant transaction, which always has been and always will be so familiar, and to others of a like nature, that I desire to call the attention of those who are interested in natural phenomena; hoping to share with them some of the delight that I have myself felt, in contemplating the exquisite forms that the camera has revealed, and in watching the progress of a multitude of events, compressed indeed within the limits of a few hundredths of a second, but none the less orderly and inevitable, and of which the sequence is in part easy to anticipate and understand, while in part it taxes the highest mathematical powers to elucidate.

In these modern days of kinematographs and snapshot cameras it might seem an easy matter to follow, by the aid of photography, even a splashing drop. But in reality the task is not so simple, for the changes of form that take place in a splash are far too rapid to come within reach of any ordinary kinematograph, and even the quickest photographic shutter is also much too slow, so that it is necessary to have recourse to the far shorter exposure of a suitable electric spark. The originals of the photographs which illustrate this book were taken by means of a spark, whose duration was certainly less than three-millionths of a second, an interval of time which bears to a whole second about the same proportion as a day to a thousand years.

In order to obtain the photographs, advantage was taken of the fact that whatever be the sequence of events in any particular splash, this sequence will be exactly repeated every time that a falling drop strikes the surface under exactly the same conditions, and the problem to be solved was, therefore, as follows:—To cause a drop of definite size to fall from a definite height in absolute darkness so as to strike the surface of the liquid into which it falls at a spot towards which is directed a photographic camera with uncovered lens, and armed with an exceptionally sensitive plate, and to illuminate the drop at the instant that it just touches the surface by a flash of such excessively short duration that no appreciable change of form can take place while the drop is illuminated.

This gives us a photograph of the earliest stage. The plate must then be removed and a fresh one substituted; a second drop, of exactly the same size, must be let fall from exactly the same place, and photographed in just the same way, but the flash must now be so timed as to take place at a slightly later stage of the splash, say, one-thousandth of a second later. The photographic plate must be then again removed and a third substituted, on which a still later stage is to be depicted, and in this way the phenomenon can be followed step by step.

By adopting this process, and not attempting to follow the same individual splash throughout, we avoid two great difficulties: (1) the necessity of shifting our photographic plate or film through a distance equal to the breadth of the whole picture every five hundredth or thousandth of a second (if we wish to obtain pictures of stages so near together as this); and (2) the difficulty of obtaining brilliant flashes of light of sufficiently short duration at these very short intervals.

For these we substitute two other difficulties: (1) that of delivering the drops exactly as required; and (2) that of timing the flash on each occasion within one or two thousandths of a second, so as to pick out the exact stage we wish to photograph.

I will now describe how these two problems have been solved.

It is easy enough to arrange for the production of small drops of almost exactly equal size. They may be allowed to fall one by one at a steady rate from the end of a fine glass tube connected to a vessel in which the liquid is maintained at a constant level, as in Fig. 1, or they may be squeezed out slowly as required by means of a syringe held in a clip as in Fig. 2. Any required number of these small drops can be caught, and allowed to run together if a larger drop is to be experimented with.

Fig. 1 Fig. 2

If the liquid used is mercury, the drops may be caught in any little glass cup such as a deeply concave watch-glass; but other liquids, such as water or milk, would wet the glass and stick to it.

If, however, the inside surface of the watch-glass be first carefully smoked in the flame of a candle, then even water or milk will roll over it without sticking, and the drop thus made up will retain a spheroidal form, and can be conveyed to the place of observation in the dark room, where it is transferred to the "dropping cup."

This consists of a similar, deep, smoked watch-glass (W)—see [Plate I]—supported on the end of a small horizontal lever, a light cylindrical rod of about the dimensions of an ordinary uncut lead pencil, pivoted about a horizontal axle near the end to which the watch-glass is attached. The other end is armed with a small light piece of iron (I) and is held in position by means of an electro-magnet (M), against the action of a spring. On cutting off the current from the electro-magnet the spring, acting as a catapult, tosses up the longer arm of the lever and thus removes the watch-glass from below the drop (D), which is left unsupported in mid-air, so that it falls from a definite fixed distance into a bowl of water placed below it, towards the surface of which the camera (C) is directed. This solves problem number one. Of course, if we wish to observe the splash of a solid sphere, there is no need to smoke the surface of the watch-glass. Indeed, the sphere may be more conveniently supported on a small ring.

Now for the production and timing of the flash. Two large Leyden jars (JJ) are provided, and charged by an electrical machine on their inner coats, one positively and one negatively. Stout wires lead from the outer coats to the dark room, and terminate in a spark-gap (S) between magnesium terminals close over the surface of the water in the bowl just mentioned. If the inner coats are now connected together, the positive and negative charges unite with a dazzling flash and a simultaneous discharge and flash takes place between the two outer coats across the spark-gap in the dark room.

This latter is the illuminating spark; we have now to time it correctly.

For this purpose it is arranged that the discharge shall be effected by means of a falling metal sphere (T) which I shall call the timing sphere, which passes between two terminals S and S connected one to the inside of one jar and one to the inside of the other. These terminals are just too far apart for a spark to leap across, till the timing sphere passes between them and thus shortens the gap; then the discharge takes place, with its accompanying flash in the dark room.

The release of the timing sphere is effected by an arrangement of lever and spring controlled by an electro-magnet exactly similar to that which releases the drop in the dark room, and the two electro-magnets are on the same electric circuit, so that the drop and timing sphere are released simultaneously. But while the drop always falls the same distance, the height through which the timing sphere has to fall before producing discharge can be adjusted at will, and to great nicety, by moving its releasing-lever up or down a vertical support with a scale attached.

If, for example, a particular stage of the splash is photographed when the timing sphere falls just four feet to the gap, then by raising its releasing-lever about two-fifths of an inch, the laws of falling bodies tell us that we shall postpone the flash by just one-thousandth of a second, and the next photograph will accordingly reveal a stage just so much later.

PLATE I

Arrangement of apparatus for photographing splashes.

E is the electrical machine.

J J are the Leyden jars whose innercoats are connected to the sparking knobs S S.

L is the lever for releasing the timing sphere T.

C is the catapult.

I is the light strip of iron held down by the electro-magnet M.

D is the drop resting on the smoked watch-glass W.

M is the electro-magnet holding down the lever against the action of thecatapult, by means of the thin strip of iron I.

C is the camera directed towards theliquid L into which the drop will fall.

S is the spark-gap between magnesiumterminals connected to the outer coats of the Leyden jars.

R is the concave mirror.

It ought still to be mentioned that to make the utmost use of the illuminating power of the spark, it is necessary to place close behind it a little concave mirror (R), by means of which a compact beam of rays, which would otherwise have been wasted, is directed to the required spot. By this addition we imitate, in miniature, the search-light of a man-of-war.

As with all experimental devices, the precision attainable with this arrangement is limited by several circumstances. In the first place, the demagnetization of the iron cores of the electro-magnets, when the current is cut off, is not truly instantaneous, and the time required depends on the strength of the magnetizing current and on the temperature of the iron, which in turn will depend on the length of time for which the current has been running. This variation would be of no importance if the two magnets were exactly alike and the springs of exactly equal strength, conditions which can be nearly but not perfectly fulfilled.

PLATE II

Photographs taken to test the accuracy of the "timing."

A more important source of uncertainty arises from the fact that the time at which the spark takes place depends partly on the magnitude of the + and - charges which have been allowed to accumulate on the discharging knobs connected to the two Leyden jars, for when these charges are larger, then the spark will be longer and will take place earlier and before the timing sphere has reached the mid-position. The charging has therefore to be carefully watched by means of the indications of a suitable electrometer, and the timing sphere must on each occasion be released when the charges have just reached the right value. But even this does not entirely suffice, for the passage of the spark depends also partly on the state of the surface of the knobs, which cannot be kept at any high degree of polish.

Still, when care is taken to keep the conditions as nearly as possible constant, neither of these sources of error is serious, and the reader can judge for himself of the accuracy of the timing from the photographs given on Plate II, in which a solid sphere was let fall in the dark room past a metre scale. The timing sphere was arranged, in the first four photographs, to illuminate it at the same stage in its fall, after a descent of thirty centimetres; if the timing had been perfect the sphere would appear on each occasion at the same mark on the scale.

It will be observed that in the first, second, and fourth photographs the falling sphere is almost accurately bisected by the long line of the three-inch mark on the right-hand edge of the scale. The greatest difference of position being just about one millimetre (as read off the left-hand scale), which would correspond to an error of about 1/2700 of a second. But the third photograph is earlier, showing the sphere 4·5 millimetres higher up, a distance which implies an error of just 1/600 of a second.

A fifth photograph was then taken, with the timing arranged so as to illuminate the sphere one centimetre higher up, and it will be seen that if we compare this with No. 3, the error is again only one millimetre. Thus Nos. 3 and 5 agree very closely, but disagree with Nos. 1, 2, and 4 by about 1/600 of a second.

The photographs themselves supply the reason. For there happens to be visible on each an (out-of-focus) image of the spark, and this image is very much the same in 1, 2 and 4, but much larger and brighter in 3 and 5, showing that the knobs were then more highly charged, which would account for the spark occurring a little too early.

But when we are watching the splash made by the fall of a liquid drop, instead of a solid sphere, there is a new and more serious source of difficulty. For the drop as it lies on the smoked glass cup is not perfectly spherical, but is flattened by its own weight, as shown in Fig. 3, and on the sudden removal of the supporting cup it oscillates between an oval form, elongated vertically, and a flattened form (see Fig. 4). These oscillations are unavoidable, and their extent will depend partly on the amount of adhesion between the smoked surface and the drop, and as this adhesion is never entirely absent and is variable, depending partly on the length of time that the drop has been lying in the cup, it follows that the drop will always receive a slight tug downwards at starting, which will be greater on some occasions than on others. On this account not only will the time taken to reach the water vary slightly, but the drop will strike it sometimes when elongated and sometimes when flattened, and the resulting splash will be affected by this circumstance.

Fig. 3 Fig. 4

The four photographs on the next page were taken in succession in order to afford the reader an opportunity of judging for himself the sort of accuracy attainable when a liquid drop was concerned.

The fall was 30 centim., and the greatest discrepancy is 4·8 millimetres, corresponding to 1/560 of a second. Thus even here the error does not amount to two-thousandths of a second.

1234

Photographs taken to test the timing of a falling drop.

With higher falls the timing sphere is moving more quickly past the discharging knobs, and the error due to a longer or shorter spark is correspondingly less, so that it appears safe to say that the accuracy of the timing was such that, when all precautions were taken, any desired stage could be picked out within two-thousandths of a second.

It is not however pretended that the precautions necessary for the most accurate timing were always taken, especially in the earlier Series of Photographs, for the main object of the experiments was to find out what happened, and only incidentally to ascertain exactly how long it took to happen, and there is no doubt that on some occasions, through the smoke-film being allowed to wear away, adhesions to the dropping cup occurred, with a corresponding disturbance of the timing, before the defect was noticed and remedied.

Fig. 5

Photograph of the edge of a rapidly whirling disc.

It remains to mention, for the sake of those interested in photography, that notwithstanding the sensitiveness of the plates and the brilliance of the illuminating spark, its duration was so short that the negatives were always "under-exposed." [C] I have mentioned that the effective duration of the spark was less than three-millionths of a second. The evidence for this is the accompanying photograph (Fig. 5), taken of a cardboard disc when rotating at a rate of fifty-three turns per second; the disc was 22 cm. in diameter, and had been roughly graduated round the edge with pen and ink. The photograph of the part that was in focus shows no perceptible blurring of the edge of the marks, and with a lens, a blurring of one-tenth of a millimetre would be easily detectable. Since the edge was moving at a rate of 36·5 metres per second (about 78 miles per hour), the time taken to traverse one-tenth of a millimetre would be rather less than three-millionths of a second. Hence we may conclude that the illumination did not last so long as this.

The weakness of the negatives was met by a prolonged development of about forty minutes in a saturated solution of eikonogen. This forbade the use of any artificial light, and all the photographic processes had to be conducted in absolute darkness. To avoid the tedium of long waiting in the dark room, a light-tight tray was constructed, in which several developing dishes could be placed, and the whole brought out into the daylight and suitably rocked. In this way ten or twelve photographs could be developed simultaneously.

It may be worth while to mention here that the bright spark given by breaking the primary circuit of an induction coil at the surface of mercury was found to be of much too long duration to be useful for the purposes of splash-photography.

FOOTNOTES:

[C] The plates which I have mostly used have been Thomas's A 1 ordinary.

CHAPTER II

THE SPLASH OF A DROP—LOW FALL

We will now turn to the photographic record itself. The first series shows the splash of a drop of water weighing ·2 of a gram, and therefore 7·36 millimetres (or rather less than one-third of an inch) in diameter, falling 40 cm. (about 16 inches) into milk mixed with water. The object of adding milk to the water was to make it more visible. The addition of milk makes, as we shall see, a little but not much difference in the general character of the splash. The scale of the figures is three-quarters of the actual size.

The number written against any figure gives, on the assumption that no unobserved error has crept in, the time in decimal parts of a second that has elapsed since the stage marked "T = 0," which is nearest to the first instant of contact. The reader will understand from what has been said that the error in any of these times may be as much as two-thousandths of a second, but is not likely to be more than that, when all precautions were taken.

It will be observed that as the drop descends into the liquid the upper portion is at first not appreciably distorted, but that a little cup or crater of liquid is thrown up round it. As the drop descends further, this crater grows wider and higher and thicker in the wall, and jets are shot out from its edge or rim. These jets are visible even in the second figure. The black marks on the inside wall of the crater are due to the lamp-black carried down with the drop from the smoked surface of the supporting cup: though in one sense a disfigurement, they serve to show by their presence that the interior of the crater is lined by the original liquid which formed the drop, and thus afford useful information as to the nature of the flow.

The crater rises with great rapidity up to Fig. 4. In Fig. 5 the walls are beginning to grow thicker, while the next three figures show the crater subsiding and widening, till in Fig. 9 it lies as a mere ring of lobes on the surface, surrounding a central hollow.

SERIES I

Milk into water (40 cm.). Scale 3/4.

1 2
T = 0 3
0·002 sec. 4
0·007 sec. 5
0·018 sec. 		6
0·031 sec. 7
0·040 sec. 8
0·050 sec. 9
0·056 sec.
1
2
T = 0
3
0·002 sec.
4
0·007 sec.
5
0·018 sec.
6
0·031 sec.
7
0·040 sec.
8
0·050 sec.
9
0·056 sec.

Fig. 10 shows the beginning of the rebound, in the rising of a central column. It will be seen that the lamp-black is now all swept to the middle, indicating that the liquid of the original drop emerges at the head of the central column. Full confirmation of this is obtained from Fig. 12, which represents the emergent column obtained when the circumstances are all the same, except that we have a drop of milk falling into water instead of water falling into milk. It will be observed that the upper part only of the column is visible, precisely because it contains nearly all the milk of the drop, while the lower part, consisting chiefly of transparent water, remains invisible.

SERIES I—(continued)

10
0·064 sec.
11
0·073 sec. 		12
13
0·093 sec.

No. 15 shows the column at its greatest height, and it should be noticed that Figs. 16 and 17 show a tendency on the part of the head of this column to split off as a separate drop.

SERIES I—(continued)

14
0·103 sec.
15
0·116 sec.
16
0·129 sec.

The column in subsiding forms a "cake" of liquid round the base. The edge of this circular cake (see Figs. 17, 18, and 19) is the first well-marked ripple spreading outwards in an ever-widening circle.

SERIES I—(continued)

17
0·153 sec.
18
0·197 sec.
19
0·217 sec.

If Fig. 19 is reached without the top of the column having separated, then the splash follows the course shown in Figs. 20a to 23a, in which it will be observed that the disappearance of the first column is very quickly followed by the rise of a secondary column very different in shape, which itself subsides again, but has not yet (in 23a) formed, as it ultimately will, a second "cake" on the top of the first. Thus the second ripple follows late after the first.

SERIES I—(continued)

Alternative (a).

20a
0·240 sec.
21a
0·242 sec.
22a
0·248 sec.
23a
0·253 sec.

If, however, the summit of the primary column succeeds in breaking off (as in Fig. 18b), or even in very nearly breaking off, then the impact of this newly-formed drop forms a second slight crater on the top of the first cake, and we have the series (18b to 24b), in which it will be observed that the rim of the secondary crater spreads rapidly outwards, so that a second well-marked circular ripple in this case quickly follows the first. The secondary column that is thrown up in Fig. 23b is very like that which emerged at a much earlier stage in the (a) series.

The photographs of this (b) series show very beautifully the manner in which the advancing edge of the ripple degenerates into smaller ripples travelling with greater speed.

SERIES I—(continued)

Alternative (b).

18b
0·214 sec.
19b
0·237 sec.
20b
0·242 sec.
21b
0·244 sec.

SERIES I—(continued)

Alternative (b).

22b
0·261 sec.
23b
0·257 sec.
24b
0·311 sec.

It will be readily understood that if the splitting off of the head of the primary column happens to take place a little earlier, or on the other hand is nearly, but not quite, complete when it descends below the surface, then subsequent configurations will differ somewhat from either of the sub-series here shown.

Since any figure photographed might belong to either sequence, the disentanglement of the two series required careful consideration and long experimenting.

The reappearance of the original drop at the head of the rebounding column, of which the explanation has been given in this chapter, is easily verified by naked-eye observation.

Let the reader when he next receives a cup of tea or coffee to which no milk has yet been added, make the simple experiment of dropping into it from a spoon, at the height of fifteen or sixteen inches above the surface, a single drop of milk. He will have no difficulty in recognizing that the column which emerges carries the white milk-drop at the top only slightly stained by the liquid into which it has fallen.

In the same way naked-eye observation reveals the crater thrown up by the entry of a big rain-drop into a pool of water. In either case what we are able to glimpse is a "stationary" stage. The rebounding column reaches a maximum height, remains poised for an instant, and then descends. The same is true of the crater. It is the relatively long duration of the moment of poise that produces on the eye a clear impression where all else is blurred by rapid change.

But there is frequently a curious illusion. We often seem to see the crater with the column standing erect in the middle of it. We know now that in reality the crater has vanished before the column appears. But the image of the crater has not time to fade before that of the column is superposed on it.

Those who are accustomed "to believe nothing that they hear and only half of what they see" may be glad to find at least the latter part of their maxim so completely justified.

CHAPTER III

PRINCIPLES INVOLVED

The reader's attention has now been directed to various features which, with certain modifications, will be found in many of the splashes that we shall examine; but so far the language used has been simply descriptive and in no way explanatory. Instead of going on to describe other splashes in the same way, and thus to accumulate a great mass of uncoördinated descriptive detail, it will be better to pause for a moment in order to become acquainted with certain principles connected with the behaviour of liquids, the application of which will go a long way towards explaining what we see going on in any splash.

The first principle to be understood is that the surface layers of any liquid behave like a uniformly stretched skin or membrane, which is always endeavouring to contract and to diminish its area. If the surface is flat, like the surface of still liquid in a bowl, this surface-tension has only the effect of exerting a small inward pull on the walls of the bowl. But if the surface is curved, with a convexity outwards, then the surface layers, on account of their tension, press the interior liquid back, and thus tend to check the growth of any protuberance; while, on the other hand, if the surface is concave outwards, then the surface-tension tends to pull the interior liquid forward, and so to diminish the concavity.

Direct evidence of this surface-tension is easy to cite. We have it in any pendent drop, such as any of those shown in the accompanying figures.

WATER.

TURPENTINE.

Pendent drops (magnified 2-1/4 times).

If we ask ourselves how it is that the liquid in the interior of one of these drops does not flow out, pressed as it is by the liquid above it, the answer is that everywhere the stretched skin presses it back. A soap-bubble too presses on the air in its interior, both the outside layers and the inside layers of the thin film being curved over the interior space. This is the reason that a soap-bubble blown on the bowl of a pipe will slowly collapse again if we remove the stem of the pipe from our mouth. The bubble drives the interior air back through the pipe. And it is easy to show that if two soap-bubbles be blown on the ends of two tubes which can be connected together by opening a tap between them, then the smaller will collapse and blow out the larger. The reason of this is that in the bubble of smaller radius the surface layers are more sharply curved, and therefore exert a greater pressure on the air within. Thus if a strap be pulled at each end with a total tension T and bent over a solid cylinder of small radius, as in Fig. 6, it is easy to see that the pressure on the surface of the part of the cylinder touched by the strap is less than if the strap be bent over an equal area on a cylinder of larger radius (Fig. 7). The tension of the surface layers of a liquid causes them to act on the liquid within, exactly as does the stretched strap on the solid in these figures. If at any place the liquid presents, as it generally does, not a cylindrical surface, but one with curvature in two directions, then the pressure corresponds to what would be produced by two straps crossing at right angles, laid one over the other, each with the curvature of the surface in its direction (Fig. 8).

Fig. 6 Fig. 7
Fig. 8 Fig. 9

We can now understand why the drop that has been lying on the watch-glass should oscillate in its descent. The sharp curvature of the edge AA of the drop (see Fig. 9) tells us that the liquid there is pushed back by the pressure of the stretched surface layers, and when the supporting glass is removed the sides of the drop move inwards, driving the liquid into the lower part, the tendency being to make the drop spherical, and so to equalize the pressure of the surface at all points. But in the process the liquid overshoots the mark, and the drop becomes elongated vertically and flattened at the sides. This causes the curvature at top and bottom to be sharper than at the sides, and on this account the back-pressure of the ends soon checks the elongation and finally reverses the flow of liquid, and the drop flattens again. As an example of the way in which a concavity of the surface is pulled out by the surface-tension may be cited the dimples made by the weight of an aquatic insect, where its feet rest on the surface without penetrating it.

This same surface-tension checks the rise of the crater, and would cause it to subside again even without the action of gravity. Thus the pressures of the sharply curved crater-edge on the liquid between the crater walls are indicated by the dotted arrows in Fig. 10, and arise from the surface-tension indicated by the full arrows. During the early part of the splash the surface-tension is more important than gravity in checking the rise of the walls. For, as the numbers show, the crater of Series I is already at about its maximum height in No. 4, i.e. about seven-thousandths of a second after first contact. In this time the fall due to gravity would be only about 1/100 of an inch. Thus if gravity had not acted the crater would only have risen about 1/100 of an inch higher. The same reasoning applies to the rise of the central column, but here the curvature at the summit is much less sharp. The numbers show that the column reaches its maximum height in about 5/100 of a second after its start in No. 10, and in this time the fall due to gravity is about half an inch, so that gravity has reduced the height by this amount.

Fig. 10

The second principle which I will now mention enables us to explain the occurrence of the jets and rays at the edge of the crater and their splitting into drops.

It was shown in 1873 by the blind Belgian philosopher, Plateau,[D] that a cylinder of liquid is not a figure of stable equilibrium if its length exceeds about 3-1/7 times its diameter. Thus a long cylindrical rod of liquid, such as Fig. 11, if it could be obtained and left for a moment to itself, would at once topple into a row of sensibly equal, equidistant drops, the number of which is expressed by a very simple law, viz. that for every 3-1/7 times the diameter there is a drop, or that the distance between the centres of the drops is equal to the circumference of the cylinder.

Fig. 11

The cause of this instability is the action of the same skin-tension that we have already spoken of. Calculation shows, and Plateau was able to confirm the calculation by experiment, that if through chance agitations lobes are formed at a nearer distance apart than 3-1/7 times the radius, with hollows between as in the accompanying Fig. 12, then the curvatures will be such as to make the skin-tension push the protuberances back and pull the hollows out. But if the protuberances occur at any greater distances apart than the length of the perimeter, then the sharper curvature of the narrower parts will drive the liquid there into the parts already wider, thus any such an initial accidental inequality of diameter will go on increasing, or the whole will topple into drops.

Fig. 12

At the last moment the drops are joined by narrow necks of liquid (Fig. 13), which themselves split up into secondary droplets (Fig. 14).

Fig. 13

Fig. 14

What we have said of a straight liquid cylinder applies also to an annulus of liquid made by bending such a cylinder into a ring. This also will spontaneously segment or topple into drops according to the same law.[E] Now the edge of the crater is practically such a ring, and it topples into a more or less regular set of protuberances, the liquid being driven from the parts between into the protuberances.

Now while the crater is rising the liquid is flowing up from below towards the rim, and the spontaneous segmentation of the rim means that channels of easier flow are created, whereby the liquid is driven into the protuberances, which thus become a series of jets. These are the jets or arms which we see at the edge of the crater. Examination with a lens of some of the craters will show that the lines of easier flow leading to a jet are often marked by streaks of lamp-black in Series I, or by streaks of milk in Series II. This explanation of the formation of the jets applies also to a similar phenomenon on a much larger scale, with which the reader will be already familiar. If he has ever watched on a still day, on a straight, slightly shelving sandy shore, the waves that have just impetus enough to curl over and break, he will have noticed that up to a certain moment the wave presents a long, smooth, horizontal cylindrical edge (see Fig. 15a) from which, at a given instant, are shot out an immense array of little jets which speedily break into foam, and at the same moment the back of the wave, hitherto smooth, is seen to be furrowed or combed (see Fig. 15b). The jets are due to the segmentation of the cylindrical rim according to Plateau's law, and the ridges between the furrows mark the lines of easier flow determined by the position of the jets.

Fig. 15b

Fig. 15a

Diagrams of a breaking wave.

The tendency of the central column of Series I to separate into two parts is only another illustration of the same instability of a liquid cylinder. The column, however, is much thicker than the jets, and its surface is therefore less sharply curved, and consequently the inward pressure of the stretched curved surface is relatively slight and the segmentation proceeds only slowly. Since this segmentation must originate in some accidental tremor, we see how it is that the summit of the column may succeed in separating off on some occasions and not on others. As a matter of fact, the height of fall for this particular splash was purposely selected, so that the column thrown up should just not succeed in dividing in order that the formation of the subsequent ripples might not be disturbed by the falling in of the drops split off. But, as the reader will have perceived, the margin allowed was not quite sufficient.

The two principles that I have now explained, viz. the principle of the skin-tension, and the principle of the instability and spontaneous segmentation of a liquid cylinder, jet, or annulus, will go far to explain much that we shall see in any splash, but it is well that the reader should realize how much has been left unexplained. Why, for example, should the crater rise so suddenly and vertically immediately round the drop as it enters? Why should the drop spread itself out as a lining over the inside of the crater, turning itself inside out, as it were, and making an inverted umbrella of itself? Why when the crater subsides should it flow inwards rather than outwards, so as to throw up such a remarkable central column?

These questions, which demand that we should trace the motion of every particle of the water back to the original impulse given by the impact of the drop, are much more difficult to answer, and can only be satisfactorily dealt with by a complicated mathematical analysis. Something, however, in the way of a general explanation will be given in a later chapter.

FOOTNOTES:

[D] Statique Expérimentale et Théorique des Liquides.

[E] See Worthington on the "Segmentation of a Liquid Annulus," Proc. Roy. Soc., No. 200, 1879.

CHAPTER IV

THE SPLASH CONTINUED

I have stated that the addition of the milk to the water made but little difference in the character of the resulting splash. It does, however, make certain differences in detail, as will be gathered from an examination of the next Series Ia, which shows the effect of letting the water-drop fall from the same height into water instead of into milk. Such a splash is difficult to photograph unless the illumination is from behind. As shown in this way, the early figures of the crater might be unintelligible to the reader had he not already studied the same crater lighted up from the side. Sometimes, though the front of the crater is hardly visible directly, yet every lobe on it can be clearly traced in the inverted image seen by reflection.

The most noticeable difference between the two splashes is perhaps the very much greater number of ripples seen with the splash in pure water. This is partly because, with the illumination behind, such ripples are more easily visible, but arises chiefly from the fact that ripples are not so readily propagated over the surface of milk on account both of its smaller surface-tension and its greater viscosity. The first appearance of outward-spreading ripples is in No. 6, just round the subsiding crater.

SERIES Ia

Water into water (40 cm. fall). Scale 9/10.

1
T = 0
2
0·004 sec.
3
0·013 sec.
4
0·018 sec.

SERIES Ia—(continued)

5
0·026 sec.
6
0·042 sec.
7
0·058 sec.
8
0·073 sec.

Since the origination of these ripples is an interesting phenomenon from a physical point of view, as throwing light on the dispersion of waves travelling with different velocities, special precautions were taken to secure the most favourable conditions, and in order to clean the surface after the arrival of each drop, which inevitably brings down a little adherent lamp-black, a continuous slow stream of fresh water was maintained which swept the contaminated surface-liquid away over the edge of the vessel.

The effect of this precaution is seen by a comparison of the photographs No. 11 and No. 11a. In the first the surface was kept quite clean in the way described; in the second it had only been cleaned by skimming it with a fine wire-gauze dish.

The beginning of the descent of the first central column seems to be marked by the appearance of a slight depression round its base, which has just not begun in No. 11a, and has just begun in No. 11, and goes on increasing in Figs. 12 and 13.

SERIES Ia—(continued)

Running water. Scale reduced to 6/10.

9
0·087 sec.
10
0·014 sec.
11
0·139 sec. 		11a
0·139 sec.

SERIES Ia—(continued)

Running water. Scale 6/10.

12
0·163 sec. 		13
0·185 sec.
14
0·207 sec.
15
0·227 sec.

The same feature marks the beginning of the descent of the secondary central column, which is still rising in Fig. 17, is just poised in Fig. 18, and thence onwards shows a gradually increasing central depression. These last four figures carry us to a rather later stage than was reached in Series I.

It should be noticed that in this Series the water-drop used was of smaller diameter than that of Series I, weighing ·13 grams as against ·2 grams. By employing the smaller drop, we diminish irregularities due to oscillations of form set up on release, for the smaller drop is more spherical when lying on the dropping cup than the larger; a few photographs taken for comparison with the full-sized drop showed, however, extremely little difference in the splashes at this height of fall.

SERIES Ia—(continued)

Still water. Scale 9/10.

16
0·247 sec.
17
0·266 sec.
18
0·294 sec.

SERIES Ia—(continued)

Still water. Scale 9/10.

19
0·285 sec.
20
0·311 sec.
21
0·321 sec.

CHAPTER V

HIGHER FALLS—BUBBLE-BUILDING

It might well be expected that the effect of increasing the height of fall of our drop to 100 cm. would be simply to emphasize the phenomena already observed, and to obtain a higher crater and a taller rebounding column. Such an expectation would be mistaken. A new phenomenon makes its appearance. The crater does indeed rise to a greater height, but its mouth closes so as to form a bubble on the surface of the liquid. If the height be not too great the closing is either incomplete or at any rate only temporary, and the bubble reopens at the top to make way for the column which rises as before from the base, but is now much thicker and hardly so high as before.

In the Series II, which is now given, the drop was of milk, 7·36 mm. in diameter, and fell 100 cm. into water.

Photographs 1 and 2—to which is added 2a, though taken under slightly different conditions—show that the drop on entering punches a sheer-walled hole, for the fine line of light seen above the level of the top of the drop in Figs. 2 and 2a marks the circular cliff-like edge of the as yet undisturbed liquid. Up the vertical sides of this circular pit the liquid of the drop is streaming. This cliff is highest and perhaps clearest in Fig. 2a.

The closing of the mouth of the crater, which is just beginning in Fig. 5, is to be explained as follows. If the crater were a simple thin-walled cylinder of liquid, it would contract under the influence of the surface-tension just as does a soap-bubble, but not so fast, since the walls have only a horizontal curvature. If the wall is thinner above than below, then the upper part will contract faster than the lower, through there being less liquid to accelerate. Now the supply of liquid is from below, and will thicken the lower part of the walls first, and thus account for the faster closing of the mouth. On the other hand, the uppermost edge of the crater is the place where the checking influence of the surface-tension on the upward flow is first felt, with the result that the edge of the rim is thickened by the influx from below, so that a more or less regular rope-like annulus is formed round the edge. Now calculation shows that such an annulus, so long as its thickness is not more than 1·61 times the thickness of the wall below, will contract quicker than the wall, and this will tend to close the crater, somewhat as a bag would be closed by the contraction of an elastic cord round the mouth. This rope-like thickening of the edge is to be seen in Figs. 5 and 7, and especially in Figs. 3 and 4 of Series III on [page 63].

SERIES II

Milk into water (100 cm. fall).

1
0·002 sec.

2a

3
0·002 sec.

2
0·002 sec.

4
0·009 sec. 		5
0·018 sec.
6
0·018 sec. 		7
0·039 sec.

The photographs 9, 10, and 11 (obtained after adding a little milk to the water in order to render it more visible) were at first very puzzling. What happens is that the bubble sometimes reopens very soon (or perhaps does not quite close) as in Fig. 9, and makes way for the column which rises from the base exactly as in the previous series. This column may be dimly seen through the walls of the bubble in Fig. 9, and No. 10 shows the column alone, the bubble having opened early and receded with great velocity, a few drops round the base being all that is left of it. Nos. 10a and 10b illustrate this reopening. In 10a the milk-drop was allowed to fall again into quite pure water, and the photograph shows very beautifully the summit of the column, with the original milk-drop at the top, emerging through the reopening mouth of the bubble; and Fig. 10b shows the same at a very slightly later stage when the bubble has completely retreated.

SERIES II—(continued)

8
0·054 sec. 		9
0·085 sec.
10
0·105 sec.
10a
0·103 sec. 		10b
0·111 sec.

In Fig. 11 the bubble has been too firmly closed to reopen, and the summit has been struck by the column within. The next figure (No. 12) shows how in such a case the emergent column becomes entangled in the liquid of the bubble when it bursts. Under the influence, however, of the surface-tension, which pushes back the protuberances and pulls out the hollows, regularity of form is soon regained. Thus Fig. 13 shows the emergent columns at a later stage after such an encounter, already much more symmetrical; and the subsequent photographs (for which a good deal of milk was added for the sake of greater visibility) show a column of uniformly sedate and respectable rotundity, betraying no traces of any youthful irregularities.

SERIES II—(continued)

11
0·105 sec.
12
0·095 sec. 		13
0·113 sec.
14
0·132 sec. 		15
0·194 sec.

SERIES II—(continued)

16
0·217 sec.
17
0·240 sec.
18
0·272 sec.
19
0·311 sec.

Series III shows the effect of still further increasing the height of fall of the water-drop (to 137 cm., or about 4 ft. 6 in.), and at the same time doubling its volume so that it now weighs ·4 gram. The crater now closes in about 18/1000 of a second, and forms a comparatively permanent bubble. The rope-like thickening of the edge, already alluded to, is well seen in Figs. 3 and 4. In its earlier stages the bubble is thick-walled, rough, and furrowed, but becomes smoother and thinner the longer it lasts, both because the liquid drains down the sides and because it becomes more uniformly distributed under the equalizing influence of the surface-tension.

SERIES III

Water-drop weighing 0·4 grams falling 137 cm. (4-1/2 feet) into milk mixed with water. Scale 1/2.

1
T = 0 		2
0·005 sec.
3
0·016 sec.
4
0·019 sec.

Such a bubble may remain long closed, as in Fig. 8, becoming every moment more delicate and exquisite, or it may open at an even earlier stage, as in Fig. 9.

There is a characteristic difference between the arms of a closing and of an opening bubble. It will be noticed that up to the moment of closing the arms slope outwards. The upper portions have been projected at an earlier stage when the mouth of the crater was wider open and the flow was either actually outwards or more nearly vertical; then as the mouth contracts the arms are left behind in the upper parts.

SERIES III—(continued)

5
0·017 sec. 		6
0·020 sec.
7
0·036 sec. 		8
0·053 sec.

In an opening bubble, on the other hand, the arms are at first vertical, and later have the very characteristic inward slope of the last figure, which is also well seen in Fig. 10a of the last series. Here the edge of the opened bubble retreats outwards and downwards, leaving the arms behind.

Such is the origin of the bubbles raised by the big drops of a thunder shower on the surface of a pool. The building of each fairy dome is accomplished in less than two-hundredths of a second, and before one-tenth of a second has elapsed the whole construction may have vanished. One can almost regret that so beautiful a process should have been so long unwatched.

To build these bubbles a large drop is essential. With a drop weighing only 0·4 of a gram, even though it fall from a height of 177 cm., there is no bubble, and the splash is almost exactly that of Series Ia. The exact time required for the closing of the bubble probably depends a good deal on the phase of oscillation of the drop at the moment of entry, and, as already observed, a big drop, which lies very flat in the dropping cup, is set vibrating more strongly on liberation than a small one.

We shall see in [Chapter VII] that the impact of a rough solid sphere, if falling from a sufficient height, produces a very exquisite bubble; in this case irregularities due to oscillation are absent, and the closing can be timed with greater precision.

SERIES III—(continued)

9
0·040 sec. 		10
0·046 sec.
11
0·070 sec.

Fig. 16

Arrangement for taking photographs below the surface of the liquid.

CHAPTER VI