Online Encyclopedia

local disturbance; but, although this has suggested the name 1 since applied to all analogous phenomena, it so happens that water-waves are far from affording the simplest instance of the process in question. In the See also:

PRESENT

present See also:

ARTICLE (from Lat. articulus, a joint)

article the See also:

PRINCIPAL

principal types of wave-See also:

motion which present themselves in physics are reviewed in the See also:

order of their complexity. Only the leading features are as a See also:

RULE

rule touched upon, the reader being referred to other articles for such developments as are of See also:

INTEREST

interest mainly from the point of view of See also:

SPECIAL

special subjects. The theory of water-waves, on the other See also:

hand, will be treated in some detail. § i. Wave-See also:

PROPAGATION

Propagation in One See also:

DIMENSION (from Lat. dimensio, a measuring)

Dimension. The simplest and most easily apprehended See also:

case of wave-motion is that of the transverse vibrations of a See also:

UNIFORM

uniform tense See also:

STRING

string. The See also:

AXIS (Lat. for " axle ")

axis of x being taken along the length of the string in its undisturbed position, we denote by y the transverse displacement at any point. This is assumed to be infinitely small; the resultant lateral force on any portion of the string is then equal to the tension (P, say) multiplied by the See also:

TOTAL

total curvature of that portion, and therefore in the case of an See also:

ELEMENT (Lat. elementum)

element Sx to Py"Sx, where the accents denote differentiations with respect to x. Equating this to pSx.5, where p is the See also:

LINE

line-See also:

DENSITY (Lat. densus, thick)

density. we have where 1 The word " wave," as a substantive, is See also:

LATE

late in See also:

ENGLISH

English, not occurring till the See also:

Bible of 1551 (See also:

SKEAT, WALTER WILLIAM (1835– )

Skeat, Etym.

Dict., 1910). The proper O. Eng. word was weep, which became wawe in M. Eng. ; it is cognate with Ger. Woge, and Is allied to " wag," to move from See also:

side to side, and is to be referred to the See also:

root wegh, to carry, See also:

Lat. vehere, Eng. " weigh," &c. The O. Eng. wafian,M.Eng. waven,to fluctuate, to waver in mind, cf. waefre, restless, is cognate with M.H.G. wabelen, to move to and fro, cf. Eng. " wabble " of which the ultimate root is seen in " See also:

WHIP

whip," and in " quaver." 5, c2y .. ..

(1) c = sl (Plp) (2) The See also:

general See also:

SOLUTION (from Lat. solvere, to loosen, dissolve)

solution of (I) was given by J. le R. d'See also:

ALEMBERT, JEAN LE ROND

Alembert in 1747; it is y = f (ct—x) +F (ct +x), (3) where the functions f, F are arbitrary. The first term is unaltered in value when x and ct are increased by equal amounts; hence this term, taken by itself, represents a wave-See also:

FORM (Lat. forma)

form which is propagated without change in the direction of x-See also:

POSITIVE (or PORTABLE) ORGAN

positive with the See also:

CONSTANT, BENJAMIN (1845-1902)

constant velocity c. The second term represents in like manner a wave-form travelling with the same velocity in the direction of x-negative; and the most general See also:

FREE

free motion of the string consists of two such wave-forms superposed. In the case of an initial disturbance See also:

con-fined to a finite portion of an unlimited string, the Motion finally resolves itself into two waves travelling unchanged in opposite directions. In these See also:

SEPARATE

separate waves we have y=Tcy', . (4) as appears fmm (3), or from See also:

SIMPLE

simple geometrical considerations. It is to be noticed, in this as in all analogous cases, that the wave-velocity appears as the square root of the ratio of two quantities, one of which represents (in a generalized sense) the See also:

ELASTICITY

elasticity of the medium, and the other its inertia. The expressions for the kinetic and potential energies of any portion of the string are T=ipfy2dx, V=aPJy,2dx, . . . (5) where the integrations extend over the portion considered. The relation (4) shows that in a single progressive wave the total See also:

ENERGY (from the Gr. ivipyela; iv, in, ipyov, work)

energy is See also:

HALF

half kinetic and half potential. \Vhen a point of the string (say the origin 0) is fixed, the solution takes the form y =-f(ct—x)f(ct-hx).

(6) As applied (for instance) to the portion of the string to the See also:

LEFT

left of 0, this indicates the superposition of a reflected wave represented by the second term on the See also:

DIRECT

direct wave represented by the first. The reflected wave has the same amplitudes at corresponding points as the incident wave, as is indeed required by the principle of energy, but its sign is reversed. The reflection of a wave at the junction of two strings of unequal densities p, p' is of interest on See also:

account of the See also:

OPTICAL

optical See also:

ANALOGY (Gr. avaXo-yLa, proportion)

analogy. If A, B be the ratios of the amplitudes in the reflected and transmitted waves, respectively, to the corresponding amplitudes in the incident wave, it is found that A=–(p–I)I(,.+1), B=2p/(p+I), . (7) where p, = (p'/p), is the ratio of the wave-velocities. This is on the See also:

HYPOTHESIS (from Gr. inrorcOivat, to put under; cf. Lat. suppositio, from sub-ponere)

hypothesis of an abrupt change of density; if the transition be gradual there may be little or no reflection. The theory of waves of See also:

LONGITUDINAL

longitudinal vibration in a uniform straight See also:

rod follows exactly the same lines. If denote the displacement of a particle whose undisturbed position is x, the length of an element of the central line is altered from Sx to Sx+SE, and the See also:

ELONGATION

elongation is therefore measured by ::'. The tension across any See also:

section is accordingly See also:

Ewe', where w is the sectional See also:

AREA

area, and E denotes See also:

Young's modulus for the material of the rod (see ELASTICITY). The See also:

RATE

rate of change. of momentum of the portion included between two consecutive See also:

CROSS

cross-sections is pwax.E, where p now stands for the See also:

VOLUME

volume-density. Equating this to the difference of the tensions on these sections we obtain c = (E/p). . (9) The solution and the See also:

INTERPRETATION (from Lat. interpretari, to expound, explain, inter pres, an agent, go-between, interpreter; inter, between, and the root pret-, possibly connected with that seen either in Greek 4 p4'ew, to speak, or irpa-rrecv, to do)

interpretation are the same as in the case of (i).

It may be noted that in an See also:

IRON [symbol Fe, atomic weight 55.85 (0=16)]

iron or See also:

STEEL, FLORA ANNIE (1847– )

steel rod the wave-velocity given by (9) amounts roughly to about five kilometres per second. The theory of See also:

PLANE

plane elastic waves in an unlimited medium, whether fluid or solid, leads to See also:

DIFFERENTIAL

differential equations of exactly the same type. Thus in the case of a fluid medium, if the displacement normal to the wave-fronts be a See also:

FUNCTION

function of t and x, only, the See also:

EQUATION (from Lat. aequatio, aequare, to equalize)

equation of motion of a thin stratum initially bounded by the planes x and x+bx is a2E pZ=—a , . (ro) where p is the pressure, and po the undisturbed density. If p depends only on the density, we may write, for small disturbances, p=po+ks, (If) where s, = (p—pu)po, is the " condensation," and k is the coefficient of cubic elasticity. Since s=—aE/ax, this leads to 02E :a 2E a12 See also:

AXE (0. Eng. aex; a word common, in different forms, in the Teutonic languages, and akin to the Greek 41v17; the New English Dictionary prefers the spelling " ax ")

axe with c = d (kip). . (13) The latter See also:

formula gives for the velocity of See also:

sound in water a value (about 1490 metres per second at 15° C.) which is in See also:

GOOD, JOHN MASON (1764-1827)

good agreement with direct observation. In the case of a See also:

gas, if we neglect See also:

variations of temperature, we have k=po by See also:

Boyle's See also:

Law, and therefore = d (pulps). This result, which is due substantially to See also:

Sir I. See also:

Newton, gives, however, a value considerably below the true velocity of sound. The discrepancy was explained by P. S.

See also:

LAPLACE, PIERRE SIMON, MARQUIS DE (1749—1827)

Laplace (about425 1806?). The temperature is not really constant, but rises and falls as the gas is alternately compressed and rarefied. When this is allowed for we have k=ypo, where 7 is the ratio of the two specific heats of the gas, and therefore c = d (ypo/po)• For See also:

air, y =1.41, and the consequent value of c agrees well with the best direct de-terminations (332 metres per second at o° C.). The potential energy of a See also:

SYSTEM

system of sound waves is iks2 per unit volume. As in all cases of propagation in one dimension, the energy of a single progressive system is half kinetic and half potential. In the case of an unlimited isotropic elastic solid medium two types of plane waves are possible, viz. the displacement may be normal or tangential to the wave-fronts. The axis of x being taken in the direction of propagation, then in the case of a normal displacement E the See also:

TRACTION (Lat. trahere, to draw)

traction normal to the wave-front is (X+2p)af/ax, where X, p are the elastic constants of the medium, viz.µ is the " rigidity," and X=k-3p, where k is the cubic elasticity. This leads to the equation E=(14) a= {(X+2a)/al =d {(k+,p)/p}. . (15) The wave-velocity is greater than in the case of the longitudinal vibrations of a rod, owing to the lateral yielding which takes See also:

PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place in the latter case. In the case of a displacement n parallel to the axis of y, and therefore tangential to the wave-fronts, we have a shearing See also:

STRAIN (through O. Fr. straindre, estraindre, mod. i treindre, from Lat. stringere, to draw tight, related to stress, stretch, string, &c.)

strain a,,/ax, and a corresponding shearing stress pap/ax. This leads to = b2,n" (16) with b = sl (a/a). . (17) In the case of steel (k=1.841 .

Io12, p=8.19. 1o", p=7.849 C.G.S.) the wave-velocities a, b come out to be 6.1 and 3.2 kilometres per second, respectively. If the medium be crystalline the velocity of propagation of plane waves will depend also on the aspect of the wave-front. For any given direction of the wave-normal there are in the most general case three distinct velocities of wave-propagation, each with its own direction of particle-vibration. These latter directions are perpendicular to each other, but in general oblique to the wave-front. For certain types of crystalline structure the results simplify, but it is unnecessary to enter into further details, as the See also:

MATTER

matter is chiefly of interest in relation to the now abandohed elastic-solid theories of See also:

double-See also:

refraction. For the See also:

MODERN

modern electric theory of See also:

LIGHT

light see LIGHT, and ELECTRIC WAVES. Finally, it may be noticed that the conditions of wave-propagation without change of type may be investigated in another manner. If we impress on the whole medium a velocity equal and opposite to that of the wave we obtain a " steady " or " stationary " state in which the circumstances at any particular point of space are constant. Thus in the case of the vibrations of an inextensible string we may, in the first instance, imagine the string to run through a fixed smooth See also:

TUBE (Lat. tuba)

tube having the form of the wave. The velocity c being constant there is no tangential See also:

ACCELERATION (from Lat. accelerare, to hasten, celer, quick)

acceleration, and the tension P is accordingly uniform. The resultant of the tensions on the two ends of an element Ss is PSs/R, in the direction of the normal, where R denotes the See also:

RADIUS

radius of curvature.

This will be exactly sufficient to produce the normal acceleration c2/R in the See also:

mass pas, provided c2 = P/p. Under this See also:

CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

condition the tube, which now exerts no pressure on the string, may be abolished, and we have a free stationary wave on a moving string. This See also:

ARGUMENT

argument is due to P. G. See also:

Tait. The method was applied to the case of air-waves by W. J. M. See also:

RANKINE, WILLIAM JOHN MACQUORN (182o-1872)

Rankine in 187o. When a gas flows steadily through a straight tube of unit section, the mass m which crosses any section in unit time must be the same; hence if u be the velocity we have pu=m (18) Again, the mass which at time t occupies the space between two fixed sections (which we will distinguish by suffixes) has its momentum increased in the time 61 by (mug—mug) 61, whence pi–P2 = m (u2—ut). (19) Combined with (18) this gives Pi +m2/pi = P2 +m2/p2 (20) Hence for absolutely steady motion it is essential that the expression p+m2/p should have the same value throughout the wave. This condition is not accurately fulfilled by any known substance, whether subject to the " isothermal " or " adiabatic " condition; but in the case of small variations of pressure and density the relation is See also:

EQUIVALENT

equivalent to m2=p2dp/dp, .

. . . (2I) and therefore by (18), if c denote the general velocity of the current, c2=dp/dp=k/p, . . . (22) in agreement with (13). The fact that the condition (20) can only be satisfied approximately shows that some progressive change of type must inevitably take place in sound-waves of finite See also:

AMPLITUDE (from Lat. amplus, large)

amplitude. This question has been examined by S. D. See also:

POISSON, SIMEON DENIS (1781-1840)

Poisson (1807), Sir G. G. See also:

Stokes (1848), B. See also:

RIEMANN, GEORG FRIEDRICH BERNHARD (1826–1866)

Riemann (1858), S. Earnshaw (1858), W.

J. M. Rankine (187o), See also:

Lord See also:

RAYLEIGH, JOHN WILLIAM STRUTT

Rayleigh (1878) and others. It appears that where (8) Ec22;", (12) where § 2. Wave-Propagation in General. We have next to consider the processes of wave-propagation in two or three dimensions. The simplest case is that of air-waves. When terms of the second order in the velocities are neglected, the dynamical equations are au ap av ap aw P°at=P°at=—ay' P°a=—az; ' • (I) and the " equation of continuity " (see See also:

HYDROMECHANICS (Gr. ubpops avuca)

HYDROMECHANICS) iS ac+Po (az+aav aw y+ az) =0. If we write p=po(I+s), p=po+ks, these may be written au as av= Zas aw ZOs at = —c2— x' at —c-- at_ - -` aZ' where c is given by § i (13), and as fau av aw at= ax+ay+az (4) the latter equation expressing that the condensation s is diminishing at a rate equal to the " divergence " of the vector (u, v, w) (see VECTOR See also:

ANALYSIS (Gr. avci and Meta, to break up into parts)

ANALYSIS). Eliminating u, v, w, we obtain a2s = c2v2s at2 where v2 stands for Laplace's operator See also:

A2(z)

a2/ax2+a2/ay2+a2/az2. This, the general equation of sound-waves, appears to be due to L. See also:

EULER, LEONHARD (1707-1783)

Euler (1759).

In the particular case where the disturbance is symmetrical with respect to a centre 0, it takes the simpler form 02(rs) =CZa2(rs) (6) at, See also:

art ' where r denotes distance from O. It is easily deduced from (1) that in the case of a medium initially at See also:

REST (O. Eng. rest, reste, bed, cognate with other Teutonic forms, e.g. Ger. Rast, Riiste, rest, and probably Gothic Rasta, league, i.e. resting or stopping place)

rest the velocity (u, v, w) is now wholly radial. The solution of (6) is s— f(ct—r)F(ct+r)' r r This represents two spherical waves travelling outwards and in-wards, respectively, with the velocity c, but there is now a progressive change of amplitude. Thus in the case of the diverging wave re-presented by the first term, the condensation in any particular part of the wave continually diminishes as I/r as the wave spreads. The potential energy per unit volume [§ r (5)1 varies as s2, and so diminishes in inverse proportion to the square of the distance from 0. It may be shown that as in the case of plane waves the total energy of a diverging (or a converging) wave is half potential and half kinetic. ' The solution of the general equation (5), first given by S. D. Poisson in 1819, expresses the value of s at any given point Pat time t, in terms of the mean values of s and I' at the instant t=o over a spherical surface of radius ct described with P as centre, viz. sp=4- f f F(ct)dui +~i[ -J Jf(ct)cko], (8) where the integrations extend over the surface of the aforesaid See also:

SPHERE (Gr. vclsa?pa, a ball or globe)

sphere, See also:

DW(i+at)X587

dw is the solid See also:

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle subtended at P by an element of its surface, and f(ct), F(ct) respectively denote the See also:

ORIGINAL

original values of s ands at the position of the element. Hence, if the disturbance be originally confined to a limited region, the agitation at any point P See also:

EXTERNAL

external to this region will begin after a time rI/c and will cease after a time See also:

R2(rt-s2)

r2/c, where r1, ri are the least and greatest distances of P. from the boundary of the region in question. The region occupied by the disturbance at any instant t is therefore delimited by the envelope of a See also:

FAMILY

family of See also:

SPHERES, MUSIC OF THE

spheres of radius ct described with the points of the original boundary as centres.

One remarkable point about waves diverging in three dimensions remains to be noticed. It easily appears from (3) that the value of 'the integral fsdt at any point P, taken over the whole time of transit of a wave, is See also:

INDEPENDENT

independent of the position of P, and therefore equal to zero, as is seen by taking P at an See also:

INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

infinite distance from the original seat of disturbance. This shows that a diverging wave necessarily contains both condensed and rarefied portions. If initially we have zero velocity everywhere, but a uniform condensation so throughout a spherical space of radius a, it is found that we have ultimately a diverging wave in the form of a spherical See also:

shell of thickness 2a, and that the value of s within this shell varies from isoa/r at the anterior See also:

FACE (from Lat. fades, derived either from facere, to make, or from a root fa-, meaning " appear "; cf. Gr. cbatvstv)

face to —isoa/r at the interior face, r denoting the mean radius of the shell. The process of wave-propagation in two dimensions offers some peculiarities which are exemplified in cylindrical waves of sound, in waves on a uniform tense plane membrane, and in See also:

ANNULAR, ANNULATE

annular waveson a See also:

HORIZONTAL

horizontal See also:

SHEET

sheet of water of (relatively) small See also:

DEPTH

depth. The equation of motion is in all these cases of the form a2s -czvl2s, at2 (9) where v12 = a2/ax2-1-a2/ay2. In the case of the membrane s denotes the displacement normal to its plane; in the application to water-waves it represents the See also:

ELEVATION

elevation of the surface above the undisturbed level. The solution of (9), even in the case of symmetry about the origin, is analytically A much less simple than that of (6). It appears that the wave due to a transient local disturbance, even of the simplest type, is now not sharply defined in the See also:

REAR

rear, as it is in the front, but has an B indefinitely prolonged "tail." This is illustrated by the annexed figures which represent graphically the time-variations in the condensation s at a particular point, as a wave originating in a local condensation passes over this point. The See also:

CURVE (Lat. curvus, bent)

curve A represents (in a typical case) the effect of a plane wave, B that of a cylindrical wave, and C that of a spherical wave. The changes of type from A to B and from B to C are accounted for by the increasing degree of mobility of the medium. The equations governing the displacements u, v, w of a uniform isotropic elastic solid medium are a2u aA P atz = (A+µ)ax +µv2u, a2v aA (IO) Patz = (X+µ)ay +µv"-v, a2w aA p atz = (X+A) az+µv2w, where A— au+av +aw ax ay az From these we derive by differentiation at2=a2v2A, .

(12) a 2 =6219,2E, ,92,7 b2v2i1, ~t2 =b2v23', (13) where aw av au aw av au E, ?b 8ya. az— 8 axay' (14) and a2=(X+2µ)/p, 1.2=See also:

Alp, . . (15) as in § 1. It appears then that the " See also:

DILATATION (from Lat. dis-, distributive, and lotus, wide)

dilatation " A and the " rotations " , r-, are propagated with the velocities a, b, respectively. By formulae analogous to (8) we can calculate the values of A, E, i, at any instant in terms of the initial conditions. The subsequent determination of u, v, w is a merely See also:

ANALYTICAL

analytical problem into which we do not enter; it is clear, however, that if the original disturbance be confined to a limited region we have ultimately two concentric spherical diverging waves. In the See also:

OUTER

outer one of these, which travels with the velocity a, the rotations l;, n, vanish, and the wave is accordingly described as irrotational," or " condensational." In the inner wave, which travels with the smaller velocity b, the dilatation A vanishes, and the wave is therefore characterized as " equivoluminal " or " distortional." In the former wave the directions of vibration of the particles tend to become normal, and in the latter tangential, to the wave-front, as in the case of plane elastic waves (§ I) The problems of reflection and transmission which arise when a wave encounters the boundary of an elastic-solid medium, or the interface of two such See also:

MEDIA

media, are of interest chiefly in relation to the older theories of See also:

OPTICS

optics. It may, however, be See also:

worth while to remark that an irrotational or an equivoluminal wave does not in general give rise to a reflected (or transmitted) wave of single See also:

CHARACTER (Gr. xapareri7p, from xap&crew, to scratch)

character; thus an equivoluminal wave gives rise to an irrotational as well as an equivoluminal reflected wave, and so on. Finally, in a limited elastic solid we may also have systems of waves of a different type. These travel over the surface with a definite velocity somewhat less than that of the equivoluminal waves above referred te; thus in an incompressible solid the velocity is •9554b; in a solid such that 71=µ it is .91941.. The agitation due to these waves is confined to the immediate neighbourhood of the surface, diminishing exponentially with increasing depth. The theory of these surface waves was given by Lord Rayleigh in 1885. In the modern theory of earthquakes three phases of the disturbance i Figures 1, 2, 4, 6, 7 and 8 are from See also:

PROFESSOR (the Latin noun formed from the verb profiteri, to declare publicly, to acknowledge, profess)

Professor See also:

HORACE [QuINTus HORATIUS FLACCUS] (65-8 B.C.)

Horace See also:

Lamb's See also:

HYDRODYNAMICS (Gr. Scot)

Hydrodynamics, by permission of the See also:

Cambridge University See also:

PRESS (through Fr. presse from Lat. pressare, frequentative of premere, to crush, squeeze, press)

Press.

the more condensed portions of the wave gain continually on the less condensed, the tendency being apparently towards the See also:

PRODUCTION (Lat. productionem, from producere, to pro-duce)

production of a discontinuity; somewhat analogous to a " See also:

BORE

bore in water-waves. Before this See also:

STAGE (Fr. 6/age; from Lat. stare, to stand)

stage can be reached, however, dissipative forces (so far ignored), such as viscosity and thermal See also:

CONDUCTION, ELECTRIC

conduction, come into See also:

PLAY

play. In See also:

PRACTICAL

practical See also:

ACOUSTICS (from the Gr. &eobav to hear)

acoustics the results are also modified by the diminution of amplitude due to spherical divergence. (2) . (3) (5) • (7) at a station distant from the origin are recognized; the first corresponds to the arrival of condensational waves, the second to that of distortional waves, and the third to that of the Rayleigh waves (see ELASTICITY). The theory of waves diverging from a centre in an unlimited crystalline medium has been investigated with a view to optical theory by G. See also:

Green (1839), A. L. See also:

CAUCHY, AUGUSTIN LOUIS, BARON (1789-1857)

Cauchy (183o), E. B. Christoffel (1877) and others. The surface which represents the wave-front consists of three sheets, each of which is propagated with its own special velocity.

It is hardly worth while to See also:

ATTEMPT (Lat. adtemptare, attentare, to try)

attempt an account here of the singularities of this surface, or of the simplifications which occur for various types of crystalline symmetry, as the subject has lost much of its physical interest now that the elastic-solid theory of light is practically abandoned. § 3. Water-Waves. Theory of " See also:

Long " Waves. The simplest type of water-waves is that in which the motion of the particles is mainly horizontal, and therefore (as will appear) sensibly the same for all particles in a See also:

VERTICAL (from Lat. vertex, highest point)

vertical line. The most conspicuous example is that of the forced oscillations produced by the See also:

ACTION

action of the See also:

sun and See also:

moon on the See also:

WATERS, TERRITORIAL

waters of the ocean, and it has therefore been proposed to designate by the term " tidal " all cases of wave-motion, whatever their See also:

scale, which have the above characteristic See also:

PROPERTY

property. Beginning with motion in two dimensions, let us suppose that the axis of x is See also:

DRAWN

drawn horizontally, and that of y vertically upwards. If we neglect the vertical acceleration, the pressure at any point will have the statical value due to the depth below the instantaneous position of the free surface, and the horizontal pressure-gradient Op/ax will therefore be independent of y. It follows that allparticles which at any instant See also:

lie in a plane perpendicular to Ox will retain this relative configuration throughout the motion. The equation of horizontal motion, on the hypothesis that the velocity (u) is in-finitely small, will be au— (It an P at - ax —gPax+ where n denotes the surface-elevation at the point x. Again, the equation of continuity, viz., ax+ay=O' . (2) gives au , au v= —Jovaxdy= —yax .

. .. (3) if the origin be taken at the bottom, the depth being assumed to be uniform. At the surface we have y=h-i-n, and v=an/at, subject to an See also:

ERROR (Lat. error, from errare, to wander, to err)

error of the second order in the disturbance. To this degree of approximation we have then an an at = —h ax' If we eliminate u between (r) and (4) we obtain a2n _ 02- at' Ox2' with c2=gh (6) The solution is as in § 1, and represents two wave-systems travelling with the constant velocity .1/(gh), which is that which would be acquired by a particle falling freely through a space equal to half the depth. Two distinct assumptions have been made in the foregoing investigation. The meaning of these is most easily understood if we consider the case of a simple-See also:

HARMONIC

harmonic See also:

TRAIN (M. Eng. trayn or trayne, derived through Fr. from Late Lat. trahinare, to drag, draw, Lat. trahere, cf. trail, trace, ultimately from the same source)

train of waves in which n=I?cosk(ct—x), u=0 See also:

cos k(ct—x), . . . (7) where k is a constant such that 2a/k is the wave-length X. The first See also:

ASSUMPTION, FEAST OF

assumption, viz. that the vertical acceleration may be neglected in comparison with the horizontal, is fulfilled if kh be small, i.e. if the wave-length be large compared with the depth. It is in this sense that the theory is regarded as applicable only to " long ." waves. The second assumption, which neglects terms of the second order in forming the equation (I), implies that the ratio n/h of the.surfaceelevation to the depth of the fluid must be small. The formulae (7) indicate also that in a progressive wave a particle moves forwards or backwards according as the water-surface above it is elevated or depressed relatively to the mean level.

It may also be proved that the expressions T = 4iphf u2dx, V = §gpf s,2dx, . . . (8) for the kinetic and potential energies per unit breadth are equal in the case of a progressive wave. It will be noticed that there is a very See also:

close See also:

correspondence between the theory of " long " water-waves and that of plane waves of sound, e.g. the ratio n/h corresponds exactly to the " condensation in the case of air-waves. The theory can be adapted, with very slight See also:

ADJUSTMENT (from late Lat. ad juxtare, derived from juxta, near, but early confounded with a supposed derivation from justus, right)

adjustment, to the case of waves propagated along a See also:

canal of any uniform section, provided the breadth, as well as the depth,be small compared with the wave-length. The principal change is that in (6) h must be understood to denote the mean depth. The theory was further extended by G. Green (1837) and by Lord Rayleigh to the case where the dimensions of the cross-section are variable. If the variation be sufficiently gradual there is no sensible reflection, a progressive wave travelling always with the velocity appropriate to the local mean depth. There is, however, a variation of amplitude; the constancy of the energy, combined with the equation of continuity, require that the elevation n in any particular , part of the wave should vary as b— § h-§, where b is the breadth of the water surface and h is the mean depth. Owing to its mathematical simplicity the theory of long waves in canals has been largely used to illustrate the dynamical theory of the tides. In the case of forced waves in a uniform canal, the equation (I) is replaced by at = —gax+X' .

• (9) where X represents the extraneous force. In the case of an See also:

EQUATORIAL

equatorial canal surrounding the See also:

earth, the disturbing action of the moon, supposed (for simplicity) to revolve in a circular See also:

ORBIT (from Lat. orbita, a track, orbis, a wheel)

orbit in the plane of the See also:

EQUATOR (Late Lat. aequator, from aequare, to make equal)

equator, is represented by X = — a S. 2 (of+Q+s), . . . . (io) where a is the earth's radius, H is the total range of the See also:

TIDE (O. Eng. lid, cf. Ger. Zeit, time or season, connected with root of Sanskrit a-diti, endless)

tide on the " See also:

EQUILIBRIUM (from the Lat. aequus, equal, and libra, a balance)

equilibrium theory," and o is the angular velocity of the moon relative to the rotating earth. The corresponding solution of the equations (4) and (9) is nI 2 H = 2c2 c (72a2 cos 2 (ot+-a See also:

Ass) ; u - - is gHoa2 cos 2 (at+a-i-e). The coefficient in the former of these equations is negative unless the ratio h/a exceed a2a/g, which is about 1/311. Hence unless the depth of our imagined canal be much greater than such depths as are actually met with in the See also:

sea the tides in it would be inverted, i.e. there would be See also:

low water beneath the moon and at the antipodal point, and high water on the See also:

meridian distant 900 from the moon. This is an instance of a familiar result in the theory of vibrations, viz. that in a forced oscillation of a See also:

BODY

body under a periodic force the phase is opposite to that of the force if the imposed frequency exceed that of the corresponding free vibration (see See also:

MECHANICS

MECHANICS). In the present case the See also:

period of the free oscillation in an equatorial canal 11,250 ft. deep would be about 30 See also:

HOURS, CANONICAL

hours. When the ratio n/h of the elevation to the depth is no longer treated as infinitely small, it is found that a progressive wave-system must undergo a continual change of type as it proceeds, even in a uniform canal. It was shown by Sir G.

B. See also:

AIRY, SIR GEORGE BIDDELL (1801-1892)

Airy (1845) that the more elevated portions of the wave travel with the greater velocities, the expression for the velocity of propagation being c(1+In/h) approximately. Hence the slopes will become continually steeper in front and more gradual behind, until a stage is reached at which the vertical acceleration is no longer negligible, and the theory ceases to apply. The process is exemplified by sea-waves See also:

RUNNING

running inwards in shallow water near the See also:

SHORE

shore. The theory of forced periodic waves of finite (as distinguished from infinitely small) . amplitude was also discussed by Airy. It has an application in tidal theory, in the explanation of " overtides " and See also:

compound tides " (see TIDE). § 4. Surface-Waves. This is the most familiar type of water-waves, but the theory is not altogether elementary. We will suppose in the first instance that the motion is in two dimensions x, y, horizontal and vertical respectively. The velocity-potential (see HYDROMECHANICS) must satisfy the equation aZop a2~ dx2+aye =o' . (I) and must make a¢/ay=o at the bottom, which is supposed to be plane and horizontal.

The pressure-equation is, if we neglect the square of the velocity, P = at —gy+ cont (2) Hence, if the origin be taken in the undisturbed surface, we may write, for the surface-elevation, n=gL-Jy=o (3) with the same approximation. We have also the geometrical condition an ail at _ ay1 y=o. The general solution of these equations is somewhat complicated, . (I) (4) • . (II) (4) and it is therefore usual to See also:

FIX, THEODORE (180o-1846)

fix See also:

ATTENTION (from Lat. ad-tendo, await, expect; the condition of being " stretched " or " tense ")

attention in the first place on the case of an infinitely extended wave-system of simple-harmonic See also:

PROFILE

profile, say n=f See also:

sin k(x—ct). .. (5) The corresponding value of tt) is , (6) cos h h) k(x—ci), =k cos s h kh where h denotes the depth; it is in fact easily verified that this satisfies (I), and makes a¢/ay=o, for y= —h, and that it fulfils the pressure-condition (3) at the free surface. The kinematic condition (4) will also be satisfied, provided ct=k tan hkh=2tan h2h, X . . (7) X denoting the wave-length 2a/k. It appears, on calculating the component velocities from (6), that the motion of each particle is elliptic-harmonic, the semi-axes of the orbit, horizontal and vertical, being cos h k(y+h) sin h k(y+h) (8) 13 sinhkh ' sinhkh .. ' ' where y refers to the mean level of the particle. The dimensions of the orbits diminish from the surface downwards.

The direction of motion of a surface-particle is forwards when it coincides with a See also:

crest, and backwards when it coincides with a trough, of the waves. When the wave-length is anything less than double the depth we have tan h kh=1, practically, and the formula (6) reduces to ¢=kcekkos k(x—ct) with cz=k=2w' (to) the same as if the depth were infinite. The orbits of the particles are now circles of radii Sek 1. When, on the other hand, X is moderately large compared with h, we have tan h kh=kh, and c=J(gh), in agreement with the preceding theory of " long " waves. These results date from G. Green (1839) and Sir G. B. Airy (1845). The energy of our simple-harmonic wave-train is, as usual, half kinetic and half potential, the total amount per unit area of the free surface being tgpp'. This is equal to the See also:

WORK

work which would be required to raise a stratum of fluid, of thickness equal to the surface-amplitude R, through a height 28. It has been assumed so far that the upper surface is free, the pressure there being uniform. We might also consider the case of waves on the See also:

COMMON

common surface of two liquids of different densities.

For wave-lengths which are less than double the depth of either liquid the formula (to) is replaced by c==2'r.p,+p„ (II) where p, p' are the densities of the See also:

LOWER

lower and upper fluids respectively. The diminution in the wave-velocity c has, as the formula indicates, a twofold cause; the potential energy of a given deformation of the common surface is diminished by the presence of the upper fluid in the ratio (p—p')/p, whilst the inertia is increased in the ratio (p+p')/p. When the two densities are very nearly equal the waves have little energy, and the oscillations of the common surface are very slow. This is easily observed in the case of See also:

PARAFFIN

paraffin oil over water. - To examine the progress, over the surface of deep water, of a disturbance whose initial character is given quite arbitrarily it would be necessary to resolve it by Founer's theorem into systems of simple-harmonic trains. Since each of these is propagated with the velocity proper to its own wave-length, as given by (to), the resulting wave-profile will continually alter its shape. The case of an initial local impulse has been studied in detail by S. D. Poisson (1816), A. Cauchy (1815) and others. At any subsequent instant the surface is occupied on either side by a train of waves of varying height and length, the wave-length increasing, and the height diminishing, with increasing distance (x) from the origin of the disturbance. The longer waves travel faster than the shorter, so that each wave is continually being drawn out in length, and its velocity of propagation therefore continually increases as it advances.

If we fix our attention on a particular point of the surface, the level there will rise and fall with increasing rapidity and in-creasing amplitude. These statements are all involved in Poisson's approximate formula II gt1 gill r p c - `cos ¢Y¢ C ) , . (12) which, however, is only valid under the condition that x is large compared with 4gtr. This shows moreover that the occurrence of a particular wave-length X is conditioned by the relation _1 /gX t V 2w' The foregoing description applies in the first instance only to the case of an initial impulse concentrated upon an infinitely narrow See also:

BAND

band of the surface. The corresponding results for the more practical case of a band of finite breadth are to be inferred by superposition. The initial stages of the disturbance at a distance x, which is large compared with the breadth b of the band, will have the same character as before, but when, owing to the continual diminution of the length of the waves emitted, X becomes comparable with or smaller than b, the parts of the disturbance which are due to the various parts of the band will no longer be approximately in the same phase, and we have a case of ' interference " in the optical sense. The result is in general that in the final stages the surface will be marked by a See also:

SERIES (a Latin word from serere, to join)

series of See also:

groups of waves of diminishing amplitude separated by bands of comparatively smooth water. The fact that the wave-velocity of a simple-harmonic train varies with the wave-length has an analogy in optics, in the propagation of light in a dispersive medium. In both cases we have a contrast with the simpler phenomena of waves on a tense string or of light-waves in vacuo, and the notion of " See also:

GROUP

group-velocity," as distinguished from wave-velocity, comes to be important. If in the above analysis of the disturbance due to a local impulse we denote by U the velocity with which the See also:

LOCUS (Lat. for " place "; in Gr. rinros)

locus of any particular wave-lengths a travels, we see from (13) that U=c. The actual fact that when a limited group of waves of approximately equal wave-length travels over relatively deep water the velocity of advance of the group as a whole is less than that of the individual waves composing it seems to have been first explicitly remarked by J. See also:

Scott See also:

Russell (1844).

If attention is concentrated on a particular wave, this is seen to progress through the group, gradually dying out as it approaches the front, whilst its former place in the group is occupied in See also:

SUCCESSION (Lat. successio, from succedere, to follow after)

succession by other waves which have come forward from the rear. General explanations, not restricted to the case of water-waves, have been given by Stokes, Rayleigh, and others. If the wave-length X be regarded as a function of x and t, we have t +Uax=o, (14) since A does not vary in the neighbourhood of a geometrical point travelling with velocity U, this being in fact the definition of U. Again, if we imagine a second geometrical point to move with the waves, we have aA aX ac dc aA at +`ax=axt`-AdA ax' • (15) the second member expressing the rate at which two consecutive wave-crests are separating from one another. Comparing (14) and (15), we have - U=c—Add. • (16) If a curve be constructed with A as See also:

ABSCISSA (from the Lat. abscissus, cut off)

abscissa and c as See also:

ORDINATE

ordinate, the group-velocity U will be represented by the intercept made by the tangent on the axis of c. This is illustrated by the annexed figure, which refers to the case of deep-water waves; the curve is a See also:

PARABOLA

parabola, and the intercept is half the ordinate, in accordance with the relation U = 4,c, already remarked. The physical importance of the motion of group-velocity was pointed out by 0. See also:

Reynolds (1877), who showed that the rate at which energy is propagated is only half that which would be required for the transport of the group as 0 a whole with the velocity c. The preceding investigations enable us to infer the effect of a pressure-disturb- ance travelling over the surface of still water with, say, a constant velocity c in the direction of x-negative. The abnormal pressure being supposed concentrated on an infinitely narrow band of the surface, the elevation +t at any point P may be regarded as due to a succession of infinitely small impulses de-livered over bands of the surface at equal infinitely See also:

SHORT, FRANCIS JOB (1857– )

short intervals of time on equidistant lines parallel to the (horizontal) axis of z. Of the wave-systems thus successively generated, those only will combine to produce a sensible effect at P which had their origin in the neighbourhood of a line Q whose position is determined by the a See also:

CONSIDERATION (from Lat. considerare, to look at closely, examine, generally taken to be from con-, and the base seen in sidus, sideris, a star, the word being supposed to be originally an astrological or astronomical term)

consideration that the phase at P is " stationary " for variations in the position of Q.

Now if t be the time which the source of disturbance has taken to travel from Q to its actual position 0, it appears from (12) that 0 the phase of the waves at P, originated at Q, is gt2/4x+}7r, where x=QP. The condition for stationary phase is therefore z=2x/t. . . (17) FIG. 3. In this differentiation, 0 and P are . to be regarded as fixed; hence x=c; and therefore OQ-=ct=2PQ. We have already seen that the wave-length at P is - such that PQ = Ut, where U is the corresponding group-velocity. Hence the • (9) . (13) P wave-length X at points to the right of 0 is uniform, being that proper to a wave-velocity c, viz. X=22rc2/g. The disturbance is therefore followed by a train of waves of approximately simple-harmonic profile, of the length indicated.

An approximate calculation shows that, except in the immediate neighbourhood of the source of disturbance, the surface-elevation is given by 2PosinR, • (18) =pc c where x is now measured from 0, and Po (=f pdx) represents the integral of the disturbing surface pressure over the (infinitely small) breadth of the band on which it acts. The case of a diffused pressure can be in- ferred by integration. The annexed figure gives a See also:

REPRESENTATION

representation of a particular case, obtained by a more 4• pressure is here sup- posed uniformly distributed over a band of breadth AB. A similar argument can be applied to the case of finite depth (h), but since the wave-velocity cannot exceed i1 (2gh) the results are modified if the velocity e of the travelling pressure exceeds this limit. There is then no train of waves generated, the disturbance of level being purely local. It hardly needs stating that the investigation applies also to the case of a stationary surface disturbance on a running stream, and that similar results follow when the disturbance consists in an equality of the bottom. In both cases we have a train of See also:

STANDING

standing waves on the down-stream side, of length corresponding to a wave-velocity equal to that of the stream. The effect of a disturbance confined to the neighbourhood of a point of the surface (of deep water) was also included in the investigations of Cauchy and Poisson already referred to. The formula analogous to (12), in the case of a local impulse, is t' g12 1 « ~4sin4, (19) where r denotes distance from the source. The interpretation is similar to that of the two-dimensional case, except that the amplitude of the annular waves diminishes outwards, as was to be expected, in a higher ratio. The effect of a pressure-point travelling in a straight line over the surface of deep water is interesting, as helping us to account in some degree for the See also:

PECULIAR

peculiar system of waves which is seen to accompany a See also:

SHIP

ship. The configuration of the wave-system is shown by means of the lines of equal phase in the annexed See also:

DIAGRAM

diagram, due to V.

W. Ekman (1906), which differs from the See also:

DRAWING

drawing origin- ally given by Lord See also:

KELVIN, WILLIAM THOMSON, BARON (1824-1907)

Kelvin (1887) in that it indicates the differ- ence of phase between the transverse and diverging waves at the common boundary of the two series. The two systems of waves are due to the fact that at any given instant there are two previous positions of the moving pressure-point which have transmitted vibrations of stationary phase to any given 5. figure. When the depth is finite the configuration is modified, and if it be less than c2/g, where c is the velocity of the disturbance, the transversal waves disappear. The investigations referred to have a bearing on the wave-resistance of See also:

SHIPS

ships. This is accounted for by the energy of the new wave-groups which are continually being started and left behind. Some experiments on See also:

TORPEDO

torpedo boats moving in shallow water have indicated a falling off in resistance due to the See also:

ABSENCE (Lat. absentia)

absence of transversal waves just referred to. For the effect of surface-tension and the theory of " ripples " see CAPILLARY ACTION. § 5. Surface-Waves of Finite Height. The foregoing results are based on the assumption that the amplitude may be treated as infinitely small. Various interesting investigations have been made in which this restriction is, more or less, abandoned, but we are far from possessing a See also:

COMPLETE

complete theory.

A system of exact equations giving a possible type of wave-motion on deep water was obtained by F. J. v. Gerstner in 1802, and rediscovered by W. J. M. Rankine in 1863. The orbits of the particles, in this type, are accurately circular, being defined by the equations x=a+k-tekbsink(a-et), y=b-k-lekbcosk(a-ct), . (I) where (a, b) is the mean position of the particle, k =2x/X ; and the wave-velocity is c = ig/k) _.I (gA/22r)). . (2)The lines of equal pressure, among which is included of course the surface-profile, are trochoidal curves. The extreme form of wave-profile is the See also:

CYCLOID (from Gr. iciidws, circle, and ennos, form)

cycloid, with the cusps turned upwards. The mathe- H--i ii ii ii ii matical elegance and simplicity of the formulae (I) are unfortunately counterbalanced by the fact that the consequent motion of the fluid elements proves to be " rotational " (see HYDROMECHANICS), and therefore not such as could be generated in a previously quiescent liquid by any system of forces applied to the surface. Sir G.

Stokes, in a,series of papers, applied himself to the determination of the possible " irrotational " wave-forms of finite height which satisfy the conditions of uniform propagation without change of type. The equation of the profile, in the case of infinite depth, is obtained in the form of a See also:

Fourier series, thus y = a cos kx+1ka2 cos 2kx +-'s k2a' cos 3kx + ..., the corresponding wave-velocity being approximately c-'V (2~\I+4X6 ' . . . where A =2a/k. The equation (3), so far as we have given the development, agrees with that of a trochoid (fig. 7). As in the case of Gerstner's waves the outline is sharper near the crests and flatter in the troughs than in the case'of the simple- FIG. 7. harmonic curve, and these features become accentuated as the ratio of the amplitude to the wave-length increases. It has been shown by Stokes that the extreme form of irrotational waves differs from that of the rotational Gerstner waves in that the crests form a See also:

blunt angle of 120°. Ac-cording to the calculations of J. H.

Additional information and Comments

There are no comments yet for this article.

» Add information or comments to this article.

Please link directly to this article:
Highlight the code below, right click, and select "copy." Then paste it into your website, email, or other HTML.

Site content, images, and layout Copyright © 2006 - Net Industries, worldwide.
Do not copy, download, transfer, or otherwise replicate the site content in whole or in part.

Links to articles and home page are always encouraged.

[back]
WAUTERS, EMILE (1848– )

[next]
WAVELLITE