waves water wire motion speed surface disturbance length pressure transverse
WAVE. By this term is commonly understood a state of disturbance which is propagated from one part of a medium to another. Thus it is euergy which passes, and not matter, - though in some cases the wave permanently displaces, usually to a small amount only, the medium through which it has passed. Currents, on the other hand, imply the passage of matter associated with energy.
The subject is one which, except in a few very simple or very special cases, has as yet been treated only by approximation even when the most formidable processes of modern mathematics have been employed, - so that this sketch, in which it is desired that as little as possible of higher mathematics should be employed, must be confined mainly to the statement of results. And the effects of viscosity, though very important, cannot be treated.
There are few branches of physics which do not present us with some forms of wave, so that the subject is a very extensive one: - tides, rollers, ripples, bores, breakers, sounds, radiations (whether luminous or obscure), telegraphic and telephonic signalling, earthquakes, the propagation of changes of surface-temperature into the earth's crust - all are forms of wave-motion. Several of these phenomena have been treated in other parts of this work, and will now be but briefly referred to ; others require more detailed notice.
When a medium is in stable equilibrium, it has no kinetic energy, and its potential energy is a minimum. Any local disturbance, therefore, in general involves a communication of energy to part of the medium, and it is usually by some form of wave-motion that this energy spreads to other parts of the medium. The mere withdrawal of a quantity of matter (as by lifting a floating body out of still water), local condensation of vapour in the air, the crushing of a hollow shell by external pressure, the change of volume resulting from an explosion, or from the sudden vaporization of a liquid - are known to all as common sources of violent wave-disturbance.
Waves may be free or forced. In the former class the disturbance is produced once for all, and is then propagated according to the nature of the medium and the form of the disturbance. Or the disturbance may be continued, provided the waves travel faster than does the centre of disturbance. In forced waves, on the other hand, the disturbing force continues to act so as to modify the propagation of the waves already produced. Thus, while a gale is blowing, the character of the water-waves is continually being modified ; when it subsides, we have regular oscillatory waves, or rollers, for the longer ones not only outstrip the shorter but are less speedily worn down by fluid friction. The huge mass of water which some steamers raise, especially when running at a high speed, is an excellent example of a forced wave. The ocean-tide is mainly a forced wave, depending on the continued action of the moon and sun ; but the tide-wave in an estuary or a tidal river is practically free, - being almost independent of moon and sun, and depending mainly upon the configuration of the channel, the rate of the current, and the tidal disturbance at the mouth.
In what follows we commence with a special case of extreme simplicity, where an exact solution is possible. This will be treated fully, partly on account of its own interest, partly because its results will be of material assistance in some of the less simple, and sometimes apparently quite different, cases which will afterwards come up for consideration.
(1) Transverse Waves on a ,S'tretched Wire. - In the article MECHANICS, § 265, it has been proved by the most elementary considerations that an inextensible but flexible rope, under uniform tension, when moving at a certain definite rate through a smooth tube of any form, exerts no pressure on the interior of the tube. In fact, the rope must press with a force T/r (where T is the tension and r the radius of curvature) on the unit of length in consequence of its tension, and with a force - p.v2/r (where is the speed, and p. the mass of unit length) in consequence of its inertia. That there may be no pressure on the tube, i.e., that it may be dispensed with, we must therefore have T to:2= 0, or v = „/T/tt. From this it follows that a disturbance of any form (of course with continuous curvature) runs along a stretched rope at this definite rate, and is unchanged during its progress. In the proof, the influence of gravity was left out of consideration, and this result may therefore be applied to the motion of a transverse disturbance along a stretched wire, such as that of a pianoforte, where the tension is very great in comparison with the weight of the wire. But the italicized word any, above, gives an excellent example of one of the most difficult parts of the whole subject, viz., the possibility of a solitary wave. This is a question upon which we cannot here enter.
If we restrict ourselves to slight disturbances only, theory points out and trial verifies that they are superposable. In fact, in the great majority of investigations which have been made with regard to waves, the disturbances have been assumed to be slight, so that we can avail ourselves of the principle of supeiposition of small motions (MEen,mcs, § 73), which is merely an application of the mathematical principle of "neglecting the second order of small quantities." The verification by trial is given at once by watching how the ring-ripples produced by two stones thrown into a pool pass through one another without any alteration ; that by observation is evident to any one who sees an object in sunlight, when the whole intervening space is full of intense wave-motion.
Returning to our wire, let us confine ourselves to a small transverse disturbance, in one plane, and try to discover what happens when the disturbance reaches one of the fixed ends of the wire. Whether a point of the wire be fixed or not does not matter, provided it do not move. In the figure below, two disturbances are shown, moving in opposite directions (and of course with equal speed). Of these, either is the perversion, as well as the inversion, of the other. When any part of the one reaches 0, the point halfway between them, so as to displace it upwards, the other contributes an exactly equal displacement downwards. Thus 0 remains permanently at rest, while the two waves pass through it without affecting one another ; and we may therefore assert that the wave A when it reaches the end of the wire is reflected as B, or rather that each part of A when it reaches 0 goes back as the corresponding part of B. B, in the same way, is seen to be reflected from 0 as A.
Now we can see what happens with a pianoforte wire. Any disturbance A is reflected from one end 0 as B, and at the other end is reflected as A again. Hence the state of the wire, whatever it may be, recurs exactly after such an interval as is required for the disturbance to travel, twice over, the length of the string.
Remembering that the displacements arc supposed to be very small, our fundamental result may now be expressed by saying that the force acting on unit length of the disturbed wire, to restore it to its undisturbed position, is T/r or px21r. Thus the ratio of the acceleration of each element to its curvature is the square of the rate of propagation of the wave. It will be shown below that this is the immediate interpretation of the differential equation of the wave-motion.
Let us express the position of a point of the wire, when undisturbed, by x its distance (say) from one end. Let y represent its transverse displacement at time t. Then, bearing in mind that the disturbance travels with a constant speed v, the nature of the motion, so far as we liave yet limited it, will be expressed by saying that the value of y, at a, at time t, will be the value of y, at x+v-r, at time t+ T j or, simply, y=f(vt – a).
For this expression, whatever be the function f, is unchanged in value, if t +T be put for t, and a+ pr for ; and no other expression possesses this special property. If there be a disturbance running the opposite way along the wire, we easily see that it will he represented by an expression of the form F(vt + x).
These disturbances are superposable, so that y=fivt –x)+F(vt+x) (1) expresses the most general state of disturbance which the wire can suffer under the limitations we have imposed. Before going farther we may use this to reproduce the results given above.
Now x=0 is one end of the wire, and there y is necessarily always zero. Hence so that the functions f and F differ only in sign. This condition, inserted in (1), gives us at once the state of matters indicated by the cut above.
If x=1 be the other end of the wire, we have the new condition The meaning of this is simply that the disturbance is periodic, the period being 21/v, the other result already obtained.
Fourier's theorem (see Hut:nos to ANALYSIS, or MECHANICS, ;3, 67) now shows that the expressionfmay be broken up into one definite series of sines and cosines, whence the usual results as to the various simple sounds which can be produced, together or separately, from a free stretched string.
It may be asked, and very naturally, How can this explanation of the nature of all possible transverse motions of a harp or pianoforte wire, as the result of sets of equal disturbances running along it with constant speed, be consistent with the appearance which it often presents of vibrating as a whole, or as a number of equal parts separated from one another by nodes which remain apparently at rest in spite of the disturbances to which, if the explanation be correct, they are constantly subjected? The answer is given at once by a consideration of the expression y (rt– x)– sin '11.(vt+x) (where i is any integer), which is a particular case of the general expression (1), limited to the circumstances of a wire of length 1. This indicates, as we have seen, two exactly similar and equal sets of simple harmonic waves running simultaneously, with equal speed, in opposite directions along the wire. By elementary trigonometry we can put the expression in the form irx fir et y= –2 sin cos 7- .
This indicates - first, that the points of the string where a has the values remain constantly at rest (these are the ends, and the i-1 equidistant nodes by which the wire is divided into i practically independent parts) ; second, that the form of the wire, at any instant, is a curve of sines, and that the ordinates of this curve increase and diminish simultaneously with a simple harmonic motion, - the wire resuming its undisturbed form at intervals of time //iv.
This discussion has been entered into for the purpose of showing, from as simple a point of view as possible, the production of a stationary or standing wave. The same principle applies to more complex cases, so that we need not revert to the question.
Recurring to the general expression for y in (1), it is clear that if we differentiate it twice with respect to t, and again twice with respect to x, the results will differ only by the factor v2 which occurs in each terns of the first. Thus d2y T d2y (2) da2 µ dx2 For D - is the acceleration, and the curvature.
(2) Longitudinal Waves in. a Wire or Rod. - If the displacements of the various parts of the wire be longitudinal instead of transverse, we may still suppose them to be represented graphically by the figure above - by laying off the longitudinal displacement of each point in a line through that point, in a definite plane, and perpendicular to the wire. In the figure a displacement to the right is represented by an upward line, and a displacement to the left by a downward line. The extremities of these lines will, in general, form a curve of continued curvature. And it is easy to see that the tangent of the inclination of the curve to the axis (i.e., its steepness at any point) represents the elongation of unit length of the wire at that point, while the curvature measures the rate at which this elongation increases per unit of length. The force required to produce the elongation bears to the elongation itself the ratio E, viz., Young's modulus. The acceleration of unit length is the change of this force per unit length, divided by p. Hence, by the italicized statement in (1), we have v= (MECHANICS, § 270). All the investigations above given apply to this case also, and their interpretations, with the necessary change of a word or two, remain as before.
Thus, according to our new interpretation of the figure, the front part of A indicates a wave of compression, its hind part one of elongation, of the wire, - the displacement of every point, however, being to the right. B is an equal and similar wave, its front being also a compression, and its rear an elongation, but in it the displacement of each point of the wire is to the left.
Hence the displacements of 0 continually compensate one another ; and thus a wave of compression is reflected from the fixed end of the wire as a wave of compression, but positive displacements are reflected as negative.
If we now consider a free rod, set into longitudinal vibration by friction, we are led to a particular case of reflection of a wave from a free end. The condition obviously that, at such a point, there can be neither compression nor elongation. To represent the reflected wave we must therefore take B of such a form that each part of it, when it meets at 0 the corresponding part of A, shall just annul the compression. On account of the smallness of the displacements, this amounts to saying that the successive parts of B must be equally inclined to the axis with the corresponding parts of A, but they must slope the other way. Thus the proper figure for this case is and the interpretation is that a wave of compression is reflected from a free end as an equal and similar wave of elongation ; but the disturbance at each point of the wire in the reflected wave is to the same side of its equilibrium position as in the incident wave.
This enables us to understand the nature of reflexion of a wave of sound from the end of an open organ pipe, as the former illustration suited the corresponding phenomenon in a closed one.
(3) Waves in a Linear System of Discrete Masses. - Suppose the wire above spoken of to be massless, or at least so thin, and of such materials, that the whole mass of it may be neglected in comparison with the masses of a system of equal pellets, which we now suppose to be attached to the wire at equal distances from one another. The weights of these pellets may be supported by a set of very long vertical strings, one attached to each, so that the arrangement is unaffected by gravity. The wire may be supposed to be stretched, as before, with a definite tension which is not affected by small transverse disturbances. We will take the case of transverse disturbances only, but it is easy to see that results of precisely the same form will be obtained for longitudinal disturbances. A moment's thought will convince the reader that there must be a limit to the frequency of the oscillations which can be transmitted along a system like this, though there was none such with the continuous wire. It is not difficult to find this limit.
Let the transverse displacement, at time I, of the nth pellet of the series, be called y,,; and let T be the tension of the wire, se the mass of a pellet, and a the distance from one pellet to the next. Then the equation of motion is obviously The result is that, if v be the speed of propagation of a disturbance in a uniform wire with the same tension and same mass, the period of the quickest simple harmonic transverse oscillation which can be freely transmitted in such a system is 7r times the time of running from one pellet to the next with speed v.
Instead of pellets on a tended wire, we might have a series of equal bar magnets, supported horizontally at proper distances from one another, in a line. The magnetic forces here take the place of the tension ; and by arranging the magnets with their like poles together, i.e., by inverting the alternate ones, we can produce the equivalent of pressure instead of tension along the series. If the magnets have each bifilar suspension, their masses will come in, as well as their moments of inertia, in the treatment of transverse disturbances.
This question is closely connected with Stokes's explanation of fluorescence (see LIGHT, VOL xiv. p. 602), for the effect of a disturbing force, of a shorter period than the limit given above, applied continuously to one of the pellets, would be to accumulate energy mainly in the immediate neighbourhood ; and this, if we suppose the disturbing force to cease, would be transmitted along the system in waves of periods equal at least to the limit. These would correspond to light of lower refrangibility than the incident, but having as characteristic a definite upper limit of refrangibility.
Such investigations, with their results, prepare us to expect that the usual mode of investigating the propagation of sound, to which we proceed, cannot be correct in the case of exceedingly high notes if the medium consist of discrete particles.
(4) Waves of Compression and Dilatation in a Fluid; Sound; Explosions. - Cousider the case of plane waves, where each layer of the medium moves perpendicularly to itself, and therefore may suffer dilatation or compression. The case is practically the same as that treated in (2) above, and can be represented by the same graphic method. For we may obviously consider only the matter contained in a rigid cylinder of unit sectional area, whose axis is parallel to the displacements. The only point of difference is in the law connecting pressure and consequent compression, and that, of course, depends upon the properties of the medium considered.
If the medium be a liquid, such as water, for instance, the compression may be taken as proportional to the pressure. Thus the acceleration on unit length of the column, multiplied by its mass (which in this case is simply the density of the medium), is equal to the increase of pressure per unit length, i.e., to the increase of condensation per unit length, multiplied by the resistance to compression, R. Thus the speed of the wave is 01/p, which is exactly analogous to the forms of (1) and (2). The density, p, of water is 62'3 lb per cubic foot, and for it R is about 20,000 atmospheres at 0° C., so that the speed of sound at that temperature is about 4700 feet per second.
That even intense differences of pressure take time to adjust themselves over very short distances in water was well shown by the damage sustained by the open copper cases of those of the " Challenger " thermometers which were crushed by pressure in the deep sea. When a strong glass shell (containing air only) is enclosed in a stout open iron tube whose length is two or three of its diameters, and is crushed by water pressure, the tube is flattened by excess of external pressure before the relief can reach the outside.
In the case of a gas, such as air, we must take the adiabatic relation between pressure and density. The pressure increases faster than, instead of at the same rate with, the density, as it would do if the gas followed Boyle's law. Thus the changes of pressure, instead of being equal to the changes of compression (multiplied by the modulus), exceed them in the proportion of the specific heat at constant pressure (K) to that at constant volume (N). Thus the speed of sound is ,/K/N.plp, where p is the pressure and p the density in the undisturbed air. The ratio of the two latter quantities, as we know, is very approximately proportional to the absolute temperature.
The questions of the gradual change of type or the dying away even of plane waves of sound, whether by reason of their form, by fluid friction, or by loss of energy due to radiation, are much too complex to be treated here.
In all ordinary simple, sounds even of very high pitch the displacements are extremely small compared with the wave length, so that the approximate solution gives the speed with considerable accuracy. And a very refined experimental test that this speed is independent of the pitch consists in listening to a rapid movement played by a good band at a great distance. But there seems to be little doubt that, under certain conditions at least, very loud sounds travel a great deal faster than ordinary sounds.
The above investigation gives the speed of sound relatively to the air. Relatively to the earth's surface, it has to be compounded with the motion of the air itself. But, as the speed of wind usually increases from the surface upwards, at least for a considerable height, the front of a sound-wave, moving with the wind, leans forward, and the sound (being propagated perpendicularly to the front) moves downwards ; if against the wind, upwards.
In the case of a disturbance in air due to a very sudden explosion, as of dynamite or as by the passage of a flash of lightning, it is probable that for seine distance from the source the motion is of a projectile character; and that part at least of the flash is clue to the heat developed by practically instantaneous and very great compression of each layer of air to which this violent motion extends.
Gravitation. and Surface-Tension Waves in Liquids. - Leaving out of consideration, as already sufficiently treated in a special article, the whole subject of TIDES, whether in oceans or in tidal rivers, there remain many different forms of water-waves all alike interesting and important. The most usual division of the free waves is into long waves, oscillatory waves, and ripples. The first two classes run by gravity, the third mainly by surface-tension (see CAPILLARY ACTION). But, while the long waves agitate the water to nearly the same amount at all depths, the chief disturbance clue to oscillatory waves or to ripples is confined to the upper layers of the water, from which it dies away with great rapidity in successive layers below. We will treat of these three forms in the order named.
Long Waves. - The first careful study of these waves was made by Scott RUSSELL (q.v.) in the course of an inquiry into traffic on canals. He arrived at the remarkable result that there is a definite speed, depending on the depth of the water, at which a horse can draw a canal-boat more easily than at any other speed, whether less or greater. And he pointed out that, when the boat moves at this speed, it agitates the water less, and therefore damages the banks less, than at any lower. This particular speed is thus, in fact, that of free propagation of the wave raised by the boat ; and, when the boat rides, as it were, on this wave, its speed is maintained with but little exertion on the part of the horse. If the boat be made to move slower, it leaves behind it an ever-lengthening procession of waves, of course at the expense of additional labour on the part of the horse.
The theory of the motion of such a wave is based on the hypothesis that all particles in a transverse section of the canal have, at the same instant, the same horizontal speed. However great this horizontal motion may be, the vertical motion of the water may be very small, for it depends on the change of horizontal speed from section to section only. In the investigation which follows, the energy of this vertical motion will be neglected (even at the surface, where it is greatest) in comparison with that of the horizontal motion. The hypothesis is proved to be well grounded by the actual observation of the motion of the water when a long wave of slight elevation or depression passes. A long box, with parallel sides of glass, partly filled with water, represents the canal ; and the wave is produced by slowly and slightly tilting the box, and at once restoring it to the horizontal position. The nature of the motion of the water is shown by particles of bran suspended in it. Such an apparatus may be usefully employed in verifying the theoretical result below, as to the connexion between the speed of the wave and the depth of the water, - observations of the passage of the crest being made with great exactness by means of a ray of light reflected from the surface of the water in a vertical plane parallel to the length of the canal. It may also be employed, by tilting it about an inclined position, for the study of the changes which take place in the wave as it passes from deeper to shallower water, or the reverse.
The statement of (1) above is immediately applicable to this question. For, if It be the (undisturbed) depth of the water, p its density, y and y' the elevations in two successive transverse sections at unit distance from one another, the difference of pressures (at the same level) in the two sections is /1(4 - y). The acceleration of a horizontal cylinder of unit section is the difference of pressures divided by p. But the whole depth is increased at each point in proportion as the thickness of a transverse slice is diminished. Hence, by the reasoning in (2) above, gP(ti– - P _ 2?/' Y v h , v2 =Pgh ; and the speed of propagation of the wave is that which a stone would acquire by falling through half the depth of the water. That the speed ought to be independent of the density of the liquid is clear from the fact that it is the weight of the disturbed portion which causes the motion, and that this (for equal waves in different liquids) changes proportionally to the mass to be moved.
Since we have made no hypothesis as to the form of the wave, our only assumptions being that the vertical motion is not only small in comparison with the depth, but inconsiderable in comparison with the horizontal motion, while the latter is the same at all depths in any one transverse section, it is clear that, under the same limitations, a wave of depression will run at the same speed as does a wave of elevation.
A solitary wave of elevation obviously carries across any fixed transverse plane a quantity of water equal to that which lies above the undisturbed level. If H be the mean height of this raised water, b the breadth of the canal (supposed rectangular), and A the length of the wave, the volume of this water is OIL But all particles in the transverse section behave alike ; and, when the wave has passed, the particles in all transverse sections have been treated alike. Hence the final result of the passage of the wave is that the whole of the water of the canal has been translated, in the direction of the wave's motion, through the space UT-11K or All/h. If the wave had been one of depression, the translation of the water would have been in the opposite direction to that of the wave's motion. Hence, when the wave consists of an elevation followed by a depression of equal volume, it leaves the water as it found it. Thus any permanent displacement of the water is due to inequality of troughs and crests.
A hint, though a very imperfect one, as to the formation of breakers on a gently sloping beach, is given by considering that in shallow water the front and rear of an ordinary surface-wave must move at different rates, the front being in shallower water than the rear and therefore allowing the rear to gain upon it.
Oscillatory Waves. - The typical example of these waves is found in what is called a "swell," or the regular rolling waves which continue to run in deep water after a storm. Their character is essentially periodic, and this feature at once enables us to select from the general integrals of the equations of non-rotatory fluid-motion the special forms which we require. The investigation may, without sensible loss of completeness for application, be still further simplified by the assumption that the disturbance is two-dimensional, i.e., that the motion is precisely the same in any two vertical planes drawn parallel to the direction in which the waves are travelling. The investigation is, unfortunately, very much more simple in an analytical than in a geometrical form.
If the axis of x be taken in tha surface of the undisturbed water, in the direction in which the waves arc travelling, and that of p vertically downwards, the equation for the velocity-potential (seo IIYDnumlicnAmes) is simply d=cp This is merely the "equation of continuity," - the condition that no liquid is generated, and none annihilated, during the motion.
The type of solution we seek, as above, is represented by y5= Y cos (tat - flit'), where Y depends on y alone. If this can be made to satisfy the equation of continuity, we may proceed to further tests and restrictions of it. Substitution leads to c/ y2 so that Y=AS "P Bg -"s, where A and B are arbitrary constants.
(a) If the depth of the water be unlimited, the value of A must be zero, for otherwise we should be dealing with disturbances which increase, without limit, as we go farther down. Hence, in this case, a particular integral of the equation, corresponding to a disturbance which can exist by itself, is (ifi= -"s cos (vil - ax) .
We will now avail ourselves of the supposition under which, as we have seen, disturbances are necessarily superposable, i.e., assume terms in B2 to be negligible. The ordinary kinetic equation (HvnnomEcHANics) then becomes C +.2)/P =10 - dt = gy + 93/BE -'w Sill (set - six).
(121-r) we have simply (it cly 0-= - ag+2a2, which is the condition that no water crosses the bounding surface. This determines a, without ambiguity, when as is given, and thus gives a relation between the period of a wave and its length, or between the period or the length and the speed of propagation. For we may write = - (vt - x) where A is the wave-length, and v the speed ; so that 2.7ff fir 7)1= - = - A A • Thus we have V-=2, r and the longer waves move faster, even when the vertical displacement is small in both. This is quite different from the result for sound-waves.
The components of the velocity of the particle of water whose mean position is x, y are Bug sin (mt - ax), parallel to x, and (41)) - Bag -"Y cos (mt - ax), parallel to y.
Bence the path is a circle whose radius is .kt the surface this is Bsitlg, as in fact we see at once by the equaion for the pressure, which gives for the form of the surface C = gn +MB sin (vat - six).
Each surface-particle is at the highest point of its circular path, and moving forwards, when the crest of the wave passes it. When the trough passes, it is at the lowest point of its circle and moving backwards. The radii of the circles diminish in geometrical progression at depths increasing in arithmetical progression. The factor is g-"Y= 5-20/A, so that at a depth of one wave-length only the disturbance is reduced to E-2" or about 1/535 of its surface-value.
From the investigation above we see that Atlantic rollers, of a wave-length of (say) 300 feet, travel at the rate of about 40 feet per second, or 27 miles an hour. But, even if they be of 40 feet height irons trough to crest (which is probably an exaggerated estimate), the utmost disturbance of a water particle at a depth of 300 feet is not quite half an inch from its mean position. This shows, in a very striking manner, what a mere surface-effect is in this way due to winds, and how the depths of the ocean are practically undisturbed by such causes.
This investigation has been carried to a second, and even to a third, approximation by Stokes, with the result that the ferns of a section of the surface is no longer the curve of sines, in which the crests and troughs are equal. The crests are steeper and higher, and the troughs wider and shallower, than the first approximation shows. Also the forward horizontal motion of each particle under the crest is no longer quite compensated for by its backward motion under the trough, so that what sailors call the "heave of the sea " is explained. The water is per manently displaced forwards by each succeeding wave. But this effect, like the whole disturbance, is greatest in the surface-layer and diminishes rapidly for each lower layer. The third approximation shows that the speed of the waves is greater than that above assigned, by a term depending on the square of the ratio of the height to the length of a wave.
(b) When the depth of the water is limited, we cannot make the simplification adopted in the last investigation.
If h be the depth of the water, our condition is that the vertical motion vanishes at that depth, and the relation between as and sa mu_ my 6 33h _ 6 -,,19A 339, E If h be regarded as infinite this gives as before m2= sip If, on the other hand, is be small compared with the wave-length, the equation approximates to 105= n'yft , or v2=gh ; and we have the formula for long waves again. Thus the expression above includes both extremes, - though, so far as long waves go, it limits them to harmonic forms of section.
The surface-section is still the curve of sines, but the paths of the individual particles are now ellipses. whose major axes are horizontal. Both axes decrease with great rapidity for particles considered at gradually increasing depths ; but the minor axes diminish faster than the major, so that the particles at the buttons oscillate in horizontal lines.
(8) Ripples. - Stokes in 1848 pointed out that the surface-tension of a liquid should be taken account of in finding the pressure at the free surface, but this seems not to have been done till 1871, when W. Thomson discussed its consequences. if T be the surface-tension, and r the radius of curvature of the (cylindrical) surface, in the case of oscillatory waves, the pressure at the free surface roust be considered as differing from that in the air by the quantity T/r (CAPILLARY AcrsoN). As T/gp is usually a small quantity, this. term will be negligible unless r is very small. If the waves be oscillatory, this means that their lengths must be very short, so that the depth of the fluid may be treated as infinite in comparison.
The curvature is practically ci2n/dx2, because 7L/1i is small; so that the term - Td2n/dx2 must be introduced into the kinetic equation along with p. The result is that qx 295 '1' or= - - .
Thus the speed is, in all cases, increased by the surface-tension; and the snore so the shorter is the wave-length. Hence, as the speed increases indefinitely with increase of wave-length when gravity alone acts, and also increases indefinitely the shorter the wave when surface-tension alone acts, there must be a 2111:71i7121111L speed, for some definite wave-length, when both causes are at work. It is easily seen that v5 is a minimum when A= Ao= 2r NiT/gp , and that the corresponding value of v2 is In the case of water the value of Ao is about 0.6S inch, and v0 is 0'76 in feet-seconds, nearly.
This slowest-moving oscillatory wave may therefore be regarded as the limit between waves proper and ripples. That ripples run faster the shorter they are is easily seen by watching the apparently rigid pattern of them which precedes a body moving uniformly through still water. The more rapid the motion the closer do the ripple-ridges approach one another. Excellent examples of ripples are produced by applying the stem of a vibrating tuning-fork to one side of a large rectangular box full of liquid. From the pitch of the note, and the wave-length of the ripples, we can snake (by the use of the above formula) an approximate determination of surface-tension, a quantity somewhat difficult to measure by statical processes.
The conditions of production of ripples by wind, or generally in a surface of separation of two fluids, each of which has any motion parallel to this surface, are given in HYDROMECHANICS.
Interference of Waves. - While the disturbances considered are so small as to be superposable, i.e., independent of one another, the effect of superposition is merely a kinematical question, and, as such, has been very fully treated under iliEcitAmcs 0§ 56-67). See also ACOUSTICS, LIGHT, and WAVE THEORY. Thus ripple-patterns, ordinary beats of musical sounds, composition of limar and solar ocean tides, diffraction, phenomena of polarized light in crystals or in transparent bodies in the magnetic field, &c., are all, in principle at least, simple kinematical consequences of superposition. But the phenomena called Tartini's beats, breakers, a bore, a jabble, and (generally) cases in which a sufficient approximation cannot be obtained by omitting powers of the displacements higher than the first, are not of this simple character.
As a single illustration, take one case of the first of these phenomena. The ,fact to be explained is that when two pure musical sounds, of frequencies 2 and 3 (say), that is, forming a "perfect fifth," are sounded together, we hear in addition to them a graver note, viz., that of which the first sound is the octave and the second the twelfth. When a resonator, carefully tuned to this graver note, is applied to the ear the note is usually not heard. Hence Helmholtz attributes its production to the fact that the drum of the ear (in consequence of the attachments of the ossicles) has different elastic properties for inward and for outward displacements.
The force tending to restore the drum from a displacement x may therefore be represented approximately by q,2 Thus, when the drum is exposed to the two 'sounds above mentioned. its equation of motion is which is the "difference-tone" referred to. This, of course, is eonnuunieated to the internal ear.
Helmholtz points out, however, that such sounds may be produced objectively, provided the interfering disturbances are sufficiently great. No one seems yet to have obtained any really accurate notion of the smallness of the disturbances of air which can be heard as sound. That they are excessively small has long been shown by many processes, but even more perfectly by the comparatively recent invention of the telephone.
Waves in an Elastic ,S'olicl. - Some of the more elementary parts of this very difficult question have been treated from a theoretical point of view in the article ELASTICITY. From an observational and experimental point of view some are treated under EARTHQUAKE. See also LIGHT, and WAVE THEORY, for the luminiferous medium appears to behave like an elastic solid.
Wares of Temperature and of Electric Potential. - In HEAT (j 78), and specially in the mathematical appendix to that article, will be found Fourier's treatment of beat-waves produced by periodic sources of various characters. It is sufficient to call attention here to the form of the equation for the linear motion of beat (which is the same as that for electricity and for diffusion), viz., dv rl icrly c dt 1.vk dx ' where v represents temperature, c specific heat, and k thermal conductivity.
dt---"`dx2 • If plane harmonic waves of the type r= X cos (rut - nx) are to be transmitted, we must have simultaneously (PX r/X.
cz - z7 &X= 0 , 2102, (rx = - mX The first gives X=AE"s+BS-"; and the second, by means of this, gives 2Kn2(AE''. - BE -")= - m(A - .
Thus A vanishes, which implies that the amplitude of the waves must continuously diminish as they progress. Also 2,cte2=sic v = cos (2x9/2/ - nx).
If the conductivity be not constant, then, even in the simple ease of k, = 7;70(1 + av), where a is small, the wave throws oil' others of inferior amplitude and of a different period.