WAVE .l It is not altogether easy to frame a definition which shall be precise and at the same time cover the various physical phenomena to which the term " wave " is commonly applied. Speaking. generally, we may say that it denotes a process in which a particular state is continually handed on without change, or with only gradual change, from one part of a medium to another. The most familiar instance is that of the waves which are observed to travel over the surface of water in consequence of a local disturbance; but, although this has suggested the name 1 since applied to all analogous phenomena, it so happens that waterwaves are far from affording the simplest instance of the process in question. In the present article the principal types of wavemotion which present themselves in physics are reviewed in the order of their complexity. Only the leading features are as a rule touched upon, the reader being referred to other articles for such developments as are of interest mainly from the point of view of special subjects. The theory of waterwaves, on the other hand, will be treated in some detail.
§ i. WavePropagation in One Dimension.
The simplest and most easily apprehended case of wavemotion is that of the transverse vibrations of a uniform tense string. The 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 total curvature of that portion, and therefore in the case of an element Sx to Py"Sx, where the accents denote differentiations with respect to x. Equating this to pSx.5, where p is the linedensity. we have
where
1 The word " wave," as a substantive, is late in English, not occurring till the Bible of 1551 (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 side to side, and is to be referred to the root wegh, to carry, 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 " whip," and in " quaver."
5, c2y .. .. (1)
c = sl (Plp) (2)
The general solution of (I) was given by J. le R. d'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 waveform which is propagated without change in the direction of xpositive with the constant velocity c. The second term represents in like manner a waveform travelling with the same velocity in the direction of xnegative; and the most general free motion of the string consists of two such waveforms superposed. In the case of an initial disturbance confined to a finite portion of an unlimited string, the Motion finally resolves itself into two waves travelling unchanged in opposite directions. In these separate waves we have
y=Tcy', . (4) as appears fmm (3), or from simple geometrical considerations. It is to be noticed, in this as in all analogous cases, that the wavevelocity appears as the square root of the ratio of two quantities, one of which represents (in a generalized sense) the 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 energy is 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(cthx). (6) As applied (for instance) to the portion of the string to the left of 0, this indicates the superposition of a reflected wave represented by the second term on the 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 account of the optical 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 wavevelocities. This is on the hypothesis of an abrupt change of density; if the transition be gradual there may be little or no reflection.
The theory of waves of longitudinal vibration in a uniform straight 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 elongation is therefore measured by ::'. The tension across any section is accordingly Ewe', where w is the sectional area, and E denotes Young's modulus for the material of the rod (see ELASTICITY). The rate of change. of momentum of the portion included between two consecutive crosssections is pwax.E, where p now stands for the volumedensity. Equating this to the difference of the tensions on these sections we obtain
c = (E/p). . (9)
The solution and the interpretation are the same as in the case of (i). It may be noted that in an iron or steel rod the wavevelocity given by (9) amounts roughly to about five kilometres per second.
The theory of plane elastic waves in an unlimited medium, whether fluid or solid, leads to differential equations of exactly the same type. Thus in the case of a fluid medium, if the displacement normal to the wavefronts be a function of t and x, only, the 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 axe with
c = d (kip). . (13) The latter formula gives for the velocity of sound in water a value (about 1490 metres per second at 15° C.) which is in good agreement with direct observation. In the case of a gas, if we neglect variations of temperature, we have k=po by Boyle's Law, and therefore = d (pulps). This result, which is due substantially to Sir I. Newton, gives, however, a value considerably below the true velocity of sound. The discrepancy was explained by P. S. 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 air, y =1.41, and the consequent value of c agrees well with the best direct determinations (332 metres per second at o° C.).
The potential energy of a 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 wavefronts. The axis of x being taken in the direction of propagation, then in the case of a normal displacement E the traction normal to the wavefront is (X+2p)af/ax, where X, p are the elastic constants of the medium, viz.µ is the " rigidity," and X=k3p, where k is the cubic elasticity. This leads to the equation
E=(14) a= {(X+2a)/al =d {(k+,p)/p}. . (15)
The wavevelocity is greater than in the case of the longitudinal vibrations of a rod, owing to the lateral yielding which takes place in the latter case. In the case of a displacement n parallel to the axis of y, and therefore tangential to the wavefronts, we have a shearing 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 wavevelocities 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 wavefront. For any given direction of the wavenormal there are in the most general case three distinct velocities of wavepropagation, each with its own direction of particlevibration. These latter directions are perpendicular to each other, but in general oblique to the wavefront. For certain types of crystalline structure the results simplify, but it is unnecessary to enter into further details, as the matter is chiefly of interest in relation to the now abandohed elasticsolid theories of doublerefraction. For the modern electric theory of light see LIGHT, and ELECTRIC WAVES.
Finally, it may be noticed that the conditions of wavepropagation 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 tube having the form of the wave. The velocity c being constant there is no tangential 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 radius of curvature. This will be exactly sufficient to produce the normal acceleration c2/R in the mass pas, provided c2 = P/p. Under this 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 argument is due to P. G. Tait.
The method was applied to the case of airwaves by W. J. M. 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 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 soundwaves of finite amplitude. This question has been examined by S. D. Poisson (1807), Sir G. G. Stokes (1848), B. Riemann (1858), S. Earnshaw (1858), W. J. M. Rankine (187o), Lord Rayleigh (1878) and others. It appears that
where (8)
Ec22;",
(12)
where
§ 2. WavePropagation in General.
We have next to consider the processes of wavepropagation in two or three dimensions. The simplest case is that of airwaves. 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 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 ANALYSIS). Eliminating u, v, w, we obtain
a2s
= c2v2s at2
where v2 stands for Laplace's operator a2/ax2+a2/ay2+a2/az2. This, the general equation of soundwaves, appears to be due to L. 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, art '
where r denotes distance from O. It is easily deduced from (1) that in the case of a medium initially at 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 inwards, respectively, with the velocity c, but there is now a progressive change of amplitude. Thus in the case of the diverging wave represented 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 sphere, dw is the solid angle subtended at P by an element of its surface,
and f(ct), F(ct) respectively denote the 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 external to this region will begin after a time rI/c and will cease after a time 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 family of 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 independent of the position of P, and therefore equal to zero, as is seen by taking P at an 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 shell of thickness 2a, and that the value of s within this shell varies from isoa/r at the anterior face to —isoa/r at the interior face, r denoting the mean radius of the shell.
The process of wavepropagation in two dimensions offers some peculiarities which are exemplified in cylindrical waves of sound, in waves on a uniform tense plane membrane, and in annular waveson a horizontal sheet of water of (relatively) small depth. The equation of motion is in all these cases of the form
a2s
czvl2s,
at2 (9)
where v12 = a2/ax21a2/ay2. In the case of the membrane s denotes the displacement normal to its plane; in the application to waterwaves it represents the 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 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 timevariations in the condensation s at a particular point, as a wave originating in a local condensation passes over this point. The 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=Alp, . . (15)
as in § 1. It appears then that the " 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 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 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 wavefront, 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 elasticsolid medium, or the interface of two such media, are of interest chiefly in relation to the older theories of optics. It may, however, be 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 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 Professor Horace Lamb's Hydrodynamics, by permission of the Cambridge University Press.
the more condensed portions of the wave gain continually on the less condensed, the tendency being apparently towards the production of a discontinuity; somewhat analogous to a " bore in waterwaves. Before this stage can be reached, however, dissipative forces (so far ignored), such as viscosity and thermal conduction, come into play. In practical 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. Green (1839), A. L. Cauchy (183o), E. B. Christoffel (1877) and others. The surface which represents the wavefront consists of three sheets, each of which is propagated with its own special velocity. It is hardly worth while to 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 elasticsolid theory of light is practically abandoned.
§ 3. WaterWaves. Theory of " Long " Waves.
The simplest type of waterwaves 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 vertical line. The most conspicuous example is that of the forced oscillations produced by the action of the sun and moon on the waters of the ocean, and it has therefore been proposed to designate by the term " tidal " all cases of wavemotion, whatever their scale, which have the above characteristic property.
Beginning with motion in two dimensions, let us suppose that the axis of x is 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 pressuregradient Op/ax will therefore be independent of y. It follows that allparticles which at any instant 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 infinitely small, will be
au— (It an
P
at  ax —gPax+
where n denotes the surfaceelevation 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=hin, and v=an/at, subject to an 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 wavesystems 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 simpleharmonic train of waves in which
n=I?cosk(ct—x), u=0 cos k(ct—x), . . . (7)
where k is a constant such that 2a/k is the wavelength X. The first assumption, viz. that the vertical acceleration may be neglected in comparison with the horizontal, is fulfilled if kh be small, i.e. if the wavelength 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 watersurface 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 close correspondence between the theory of " long " waterwaves and that of plane waves of sound, e.g. the ratio n/h corresponds exactly to the " condensation in the case of airwaves. The theory can be adapted, with very slight adjustment, to the case of waves propagated along a canal of any uniform section, provided the breadth, as well as the depth,be small compared with the wavelength. 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 crosssection 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 equatorial canal surrounding the earth, the disturbing action of the moon, supposed (for simplicity) to revolve in a circular orbit in the plane of the 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 tide on the " 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 Ass) ; u   is gHoa2 cos 2 (at+aie).
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 sea the tides in it would be inverted, i.e. there would be low water beneath the moon and at the antipodal point, and high water on the 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 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 MECHANICS). In the present case the period of the free oscillation in an equatorial canal 11,250 ft. deep would be about 30 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 wavesystem must undergo a continual change of type as it proceeds, even in a uniform canal. It was shown by Sir G. B. 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 seawaves running inwards in shallow water near the 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 compound tides " (see TIDE).
§ 4. SurfaceWaves.
This is the most familiar type of waterwaves, 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 velocitypotential (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 pressureequation 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 surfaceelevation,
n=gLJy=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 fix attention in the first place on the case of an infinitely extended wavesystem of simpleharmonic profile, say
n=f 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 pressurecondition (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 wavelength 2a/k. It appears, on calculating the component velocities from (6), that the motion of each particle is ellipticharmonic, the semiaxes 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 surfaceparticle is forwards when it coincides with a crest, and backwards when it coincides with a trough, of the waves.
When the wavelength 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 simpleharmonic wavetrain 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 work which would be required to raise a stratum of fluid, of thickness equal to the surfaceamplitude 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 common surface of two liquids of different densities. For wavelengths 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 lower and upper fluids respectively. The diminution in the wavevelocity 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 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 simpleharmonic trains. Since each of these is propagated with the velocity proper to its own wavelength, as given by (to), the resulting waveprofile 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 wavelength 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 increasing 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 wavelength 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
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 series of groups of waves of diminishing amplitude separated by bands of comparatively smooth water.
The fact that the wavevelocity of a simpleharmonic train varies with the wavelength 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 lightwaves in vacuo, and the notion of " groupvelocity," as distinguished from wavevelocity, 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 locus of any particular wavelengths a travels, we see from (13) that U=c. The actual fact that when a limited group of waves of approximately equal wavelength 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. Scott 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 succession by other waves which have come forward from the rear. General explanations, not restricted to the case of waterwaves, have been given by Stokes, Rayleigh, and others. If the wavelength 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 wavecrests are separating from one another. Comparing (14) and (15), we have 
U=c—Add. • (16)
If a curve be constructed with A as abscissa and c as ordinate, the groupvelocity 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 deepwater waves; the curve is a 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 groupvelocity was pointed out by 0. 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 pressuredisturb
ance travelling over the surface of still water with, say, a constant velocity c in the direction of xnegative. 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 delivered over bands of the surface at equal infinitely short intervals of time on equidistant lines parallel to the (horizontal) axis of z. Of the wavesystems 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 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 wavelength at P is  such that PQ = Ut, where U is the corresponding groupvelocity. Hence the
• (9)
. (13)
P
wavelength X at points to the right of 0 is uniform, being that proper to a wavevelocity c, viz. X=22rc2/g. The disturbance is therefore followed by a train of waves of approximately simpleharmonic profile, of the length indicated. An approximate calculation shows that, except in the immediate neighbourhood of the source of disturbance, the surfaceelevation 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 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 wavevelocity 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 standing waves on the downstream side, of length corresponding to a wavevelocity 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 twodimensional case, except that the amplitude of the annular waves diminishes outwards, as was to be expected, in a higher ratio.
The effect of a pressurepoint travelling in a straight line over
the surface of deep water is interesting, as helping us to account
in some degree for the peculiar system of waves which is seen to
accompany a ship. The configuration of the wavesystem is shown
by means of the lines of equal phase in the annexed diagram, due to
V. W. Ekman (1906), which
differs from the drawing origin
ally given by Lord 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 pressurepoint 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 waveresistance of ships. This is accounted for by the energy of the new wavegroups which are continually being started and left behind. Some experiments on torpedo boats moving in shallow water have indicated a falling off in resistance due to the absence of transversal waves just referred to. For the effect of surfacetension and the theory of " ripples " see CAPILLARY ACTION.
§ 5. SurfaceWaves 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 complete theory.
A system of exact equations giving a possible type of wavemotion 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+ktekbsink(aet), y=bklekbcosk(act), . (I)
where (a, b) is the mean position of the particle, k =2x/X ; and the wavevelocity is
c = ig/k) _.I (gA/22r)). . (2)The lines of equal pressure, among which is included of course the surfaceprofile, are trochoidal curves. The extreme form of waveprofile is the cycloid, with the cusps turned upwards. The mathe
Hi 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 " waveforms 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 Fourier series, thus
y = a cos kx+1ka2 cos 2kx +'s k2a' cos 3kx + ..., the corresponding wavevelocity 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 wavelength 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 blunt angle of 120°. According to the calculations of J. H. Michell (1893), the height is then about oneseventh of the wavelength, and the wavevelocity exceeds that of very low waves of the same length in the ratio 6:5. It is to be noticed further that in these waves of permanent type the motion of the waterparticles is not purely oscillatory, there being on the whole a gradual drift at the surface in the direction of propagation. These various conclusions appear to agree in a general way with what is observed in the case of seawaves.
In the case of finite depth the calculations are more difficult, and we can only here notice the limiting type which is obtained when the wavelength is
supposed very great compared with the depth (h). We have then practically the
" solitary wave " to which attention was first directed by J. Scott Russell (1844)
from observation. The theory has been worked out by J. Boussinesq (1871) and Lord Rayleigh. The surfaceelevation is given by
n = a sec 112 z (x/b) , . . . . (5)
b2 =h2(h+a)/3a, . (6)
and the velocity of propagation is
c= I g(h+a)} (7)
In the extreme form a=h and the crest forms an angle of 12o°. It appears that a solitary wave of depression, of permanent type, is impossible.
Mem. sur la theorie des ondes," Mem. de l'acad. roy. des sc. i (1827) ; Sir G. B. Airy, " Tides and Waves," Encycl. Metrop. (1845). Many classical investigations are now most conveniently accessible
Al Ws V. Walfrid Ekman, On Stationary Waves in Running Water.
• (3) . (4)
Y

0
s
provided
in the following collections: G. Green, Math. Papers (Cambridge, 1871); H. v. Helmholtz, Gesammelte Abhandlungen (Leipzig, 18821895); Lord Rayleigh, Scientific Papers (Cambridge, 1899—1903) ; W. J. M. Rankine, Misc. Scientific Papers (London, 1881); Sir G. G. Stokes, Math. and Phys. Papers (Cambridge, 1880—1905). Numerous references to other writers will be found in the articles by P. Forchheimer (" Hydraulik "), H. Lamb (" Schwingungen elastischer Korper, insb. Akustik "), and A. E. H. Love (" Hydrodynamik ") in various divisions of the fourth volume of the Encykl. d. math. Wiss. ; and in H. Lamb's Hydrodynamics (3rd ed., Cambridge, 1906). (H. LB.)
End of Article: WAVE 

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