# 1911 Encyclopædia Britannica/Wave

See also Wave on Wikipedia; and our 1911 Encyclopædia Britannica disclaimer.
​WAVE. 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 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 present article the principal types of wave-motion 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 water-waves, on the other hand, will be treated in some detail.
§1. Wave-Propagation in One Dimension.
The simplest and most easily apprehended case of wave-motion 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 δx to $Py^{\prime \prime }\delta x$, where the accents denote differentiations with respect to x. Equating this to $\rho \delta x.{\ddot {y}}$, where ρ is the line-density, we have
| | ${\ddot {y}}=c^{2}y^{\prime \prime },$ | (1) |
| where | $c={\sqrt}(P/\rho ).$ | (2) |
| $y=f(ct+x)+F(ct+x),\,$ | (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-form which is propagated without change in the direction of x-positive with the 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 free motion of the string consists of two such wave-forms 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
| $y=\mp cy^{\prime },\,$ | (4) |
as appears from (3), or from 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 elasticity of the medians, and the other its inertia.
The expressions for the kinetic and potential energies of any portion of the string are
| $T={\tfrac {1}{2}}\rho \int y^{2}dx,\quad V={\tfrac {1}{2}}P\int y^{\prime 2}dx$ | (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.
When a point of the string (say the origin O) is fixed, the solution takes the form
| $y=f(ct-x)-f(ct+x).\,$ | (6) |
As applied (for instance) to the portion of the string to the left of O, 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 ρ, ρ' is of interest on account of the optical analogy. If A, B be the ratios of the amplitudes us the reflected and transmitted waves, respectively, to the corresponding amplitudes in the incident wave, it is found that
| $A=-(\mu -1)/(\mu +1),\quad B=2\mu /(\mu +1),\,$ | (7) |
where μ=√(ρ′/ρ), is the ratio of the wave-velocities. 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 ξ denote the displacement of a particle whose undisturbed position is x, the length of an element of the central line is altered from δx to δx+δξ, and the elongation is therefore measured by ξ'. The tension across any section is accordingly Eωξ', where ω is 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 cross-sections is ρωδx.ξ, where ρ now stands for the volume-density. Equating this to the difference of the tensions on these sections we obtain
| $\xi =c^{2}\xi ^{\prime \prime }$ | (8) |
where
| $c={\sqrt}(E/\rho ).\,$ | (9) |
The solution and the interpretation are the same as in the case of (1). It may be noted that in an iron or steel rod the wave-velocity 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 wave-fronts be a function of t and x, only, the equation of motion of a thin stratum initially bounded by the planes x and x+δx is
| $\rho _{0}{\frac {\partial ^{2}\xi }{\partial t^{2}}}=-{\frac {\partial p}{\partial x}},$ | (10) |
where p is the pressure, and ρ$_0$ the undisturbed density. If p depends only on the density, we may write, for small disturbances,
| $p=p_{0}+ks,\,$ | (11) |
where s=(ρ−ρ$_0$)ρ$_0$, is the "condensation," and k is the coefficient of cubic elasticity. Since s=−d ξ/dx, this leads to
| ${\frac {\partial ^{2}\xi }{\partial t^{2}}}=c^{2}{\frac {\partial ^{2}\xi }{\partial x^{2}}}\,$ | (12) |
with
| $c={\sqrt}(k/p)\,$ | (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=p$_0$ by Boyle’s Law, and therefore c=√p$_0$ / ρ$_0$. 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 (about 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=γp$_0$, where γ is the ratio of the two specific heats uf the gas, and therefore c=√(γp$_0$ / ρ$_0$). For air, γ=1·41, and the consequent value of c agrees well with the best direct determinations (332 metres per second at 0° C.).
The potential energy of a system of sound waves is 1/2ks$^2$ 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 ξ the traction normal to the wave-front is (λ+2μ)∂ξ/∂x, where λ, μ are the elastic constants of the medium, viz. pi is the “rigidity,” and λ＝k2/3μ, where k is the cubic elasticity. This leads to the equation
| ξ＝a$^2$ξ″ | (14) |
| a＝√(λ+2μ)/ρ}＝√{k+4/3μ)/ρ} | (15) |
The wave-velocity 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 η parallel to the axis of y, and therefore tangential to the wave-fronts, we have a shearing strain ∂η/∂x, and a corresponding shearing stress μ∂η/∂x. This leads to
| η..＝b$^2$η | (16) |
with
| b＝√(μ/ρ). | (17) |
In the case of steel (k＝1·841 . 10$^{12}$, μ = 8·19. 10$^{11}$, ρ = 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 matter is chiefly of interest in relation to the now abandoned elastic-solid theories of double-refraction. For the modern electric theory of 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 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 δs is Pδs/R, in the direction of the normal, where R denotes the radius of curvature. This will be exactly sufficient to produce the normal acceleration c$^2$R in the mass ρδς, provided c$^2$＝P/ρ. 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 air-waves by W. J. M. Rankine in 1870. 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
| ρu＝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 δt by (mu$_2$−mu$_2$) U, whence
| p$_1$−p$_2$＝m(u$_2$−u$_1$) | (19) |
Combined with (18) this gives
| p$_1$+m$^2$/ρ$_1$＝p$_2$+m$_2$/ρ$_2$ | (20) |
Hence for absolutely steady motion it is essential that the expression p+m$^2$/ρ 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 rase of small variations of pressure and density the relation is equivalent to
| m$^2$＝ρ$^2$dp/dρ | (21) |
and therefore by (18), if c denote the general velocity of the current,
| c$^2$＝dp/dρ＝k/ρ, | (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 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 (1870), Lord Rayleigh (1878) and others. It appears that ​៛•¢¶ 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 water-waves. 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. 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 dtl dp dv dp dw dp ^at-~di- ""Jt-'Ty' ""Tt^'ai• and the " equation of continuity " (see Hydromechanics) is (I) dp, (du dv dw
(2) If we write p=po(i +s), p = pD+ks, these may be written aM ^ds dv jflj dw j3£ dt ^ dx' dl~ ^ dy' dt~ ^ dz
where c is given by § i (13), and dt^ dx^dy'^ dz)'• • the latter equation expressing that the condensation i is diminishing at a rate equal to the " divergence " of the vector (w, v, w) (see Vector Analysis). Eliminating u, v, w, we obtain (3) (4) d?='^ •
(5) where v* stands for Laplace's operator d'ldx'+d'ldy'+d-ldz^ This, the general equation of sound-waves, appears to be due to L. Euler (1759). In the particular case where the disturbance is symmetrical with respect to a centre O, it takes the simpler form ^-..^
(6) dp dr where r denotes distance from O It is easily deduced from (i) that in the case of a medium initially at rest the velocity (a, v, w) is now wholly radial. The soludon of (6) is fjct-r) . F(cl+r) r "* r
(7) 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 [§ I (5)) varies as i^ and so diminishes in inverse proportion to the square of the distance from O. 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 18 19, expresses the value of 5 at any given point P at time /, in terms of the mean values of s and s at the instant / = over a spherical surface of radius ct described with P as centre, viz. $s_{p}={\frac {t}{4\pi }}\int \int \mathrm {F} (ct)d\omega +{\frac {d}{dt}}\left[{\frac {t}{4\pi }}+\int \int f(ct)d\omega \right]$ (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(cO respectively denote the original values of s and i at the position of the element. Hence, if the disturbance be originally confined to a Hmited region, the agitation at any point P external to this region will begin after a time r, /c and will cease after a time r^fc, where ri, r^ 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 I is therefore delimited by the envelope of a family of spheres of radius cl 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 'alue 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 j within this shell varies from isoa/r at the anterior face to —^Soa/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 annular waves The on a horizontal sheet of water of ^relatively) small depth, equauon of motioa is in all these cases of the form § p=c'Vi% . . •
(9) where Vi' = 3V3x»-f a=y9y'-In the case of the membrane s denotes the displacement normal to its plane; in the application to watei-waves it represents the elevation of the surface above the undisturbed level. The sol- ution of (9), even in the case of symmetry about the origin, is .analytically 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 indefinitely prolonged "tail." This is illustrated by the annexed figures which represent graphically the time-variations in the condensation i 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 «, v, w oi a. uniform isotropic elastic solid medium are where dhv
dA,
A- —1— -I- — dx'^dy^ dz (10) (II) From these we derive by differentiation ^=aVA (12) where and g=Wl. g = 6=V», . |y = 6'v'f. «. -?. f- a-i t» ​ 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 (1830), E. B . Chrisroffel (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 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 "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 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 wave-motion, 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 pressure-gradient dpidx will therefore be independent of y. It follows that all particles 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 (m) is infinitely small, will be
du
dp
dit
^= -51= -^'^
where rj denotes the surface-elevation at the point *. Again, the
equation of continuity, viz..
du1dv
iu+a-y=°
(2)
gives
C'du,
du
if the origin be taken at the bottom, the depth being assumed to be uniform. At the surface we have y = h+i], and v = dn/dt, subject to an error of the second order in the disturbance. To this degree of approximation we have then
(4)
at
"dx-
(5)
If we eliminate u between (i) and (4) we obtain ap~'^dx
with
c'=gh
(6)
The solution is as in § I, and represents two wave-systems travelling with the constant velocity V(g/i). 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-harmonic train of waves in which V=0 cosset- x), u=^cosk(, ct-x), (7)
where fe is a constant such that 2!r/;fe is the wave-length 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 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 surface elevation to the depth of the fluid must be small. The formulae (7) indicate also that Ln 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 = hphfu'-dx, V = gf>Jrfdx,
(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 " water-waves and that of plane waves of sound, e.g. the ratio nlh corresponds exactly to the " condensation "
in the case of air-waves.
The theory can be adapted, with very slight adjustment, to the case of waves propagated along a canal of imy uniform section, provided the breadth, as well as the depth. be srnall compared with the wave-length. The principal change is that in (6) h must be understood to denote the mean depth. The theor); was further e;<tended 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 17 in any particular part of the wave should vary as b-ih-i, where 6 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 (l) is replaced by
du
dij, ..
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 = -^sin2(a<-t -f-|-0,
(10)
where a is the earth's radius, H is the total range of the tide on the " equilibrium theory, " and a is the angular velocity of the moon relative to the rotating earth. The corresponding solution of the equations (4) and (9) is
iHira
X.
(n)
The coefficient in the former of these equations is negative unless the ratio h/a exceed a^a/g, which is about 1/31 1. 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 90° 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 j)//i 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 . Airy (1845) that the more elevated portions of the wave travel with the greater velocities, the expression for the velocity of propagation being 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 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 " over tides " and " compound tides " (see Tide).
§ 4. Surface-Waves.
This is the most familieir 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 HYDROiiECHANics) must satisfy the equation
(0
and must make d<t>/dy=o at the bottom, which is supposed to be plane and horizontal. The pressure-equation is, if we neglect the square of the velocity,
37 -«>'+ const.
(2)
Hence, if the origin be taken in the undisturbed surface, we may write, for the surface-elevation, "•m, ..
gL3i.
(3)
with the same approximation.
We have also the geometrical
condition
dr,
d±
dl° ldyy^o
(4)
The general solution of these equations is somewhat complicated. ​ and it is therefore usual to fix attention in the first place on the case of an infinitely extended wave-system of simple-harmonic profile, say
| η＝β sin k(x−ct) | (5) |
The corresponding value of φ is
φ＝ cos h k{y+h)
t'-Tc
cos ha ^osHx-ct).
(6)
where h denotes the depth,
it is in fact easily verified that this satisfies (i), and makesit>/dy = 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 <;2=|tanh«; = |^tanh^. ...
(7)
X denoting the wave-length 2;r/i. It appears, on calculating the
component velocities (rom (6), that the motion of each particle is elliptic-harmonic, the semi-axes of the orbit, horizontal and vertical, being
„cosh k{y- -h) sin h k[y+h)
P sinhkli
P 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 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 kli = l, practically, and the formula (6) reduces to ^=|?e'i'cos kix—ct)
with
kc
(9)
(10)
the same as if the depth were infinite. The orbits of the particles are now circles of radii /3e'=". When, on the other hand, X is moderately large compared with h, we have tan h kh = kh, and c = -l{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 gp^.
This is equal to the work which would be required to raise a stratum of fluid, of thiclcness equal to the surface amplitude (3, through a height .
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 wave-lengths which are less than double the depth of either liquid the formula (10) is replaced by 2 gX p-p
2ir
p+p"
(n)
where p, p' are the densities of the 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')lp, whilst the inertia is increased in the ratio {p+p')lp. 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 Fourier'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 (10), 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 var^'ing height and length, the wave-length increasing, and the height diminishing, with increasing distance (ar) 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
Vx— I cos
xi
4-^
which, however, is only valid under the condition that x is large compared with ^gP. This shows moreover that the occurrence of a particular wave-length X is conditioned by the relation 7=Vl^
03)
The foregoing description applies in the first instance only to the case of aa 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 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 m vacuo, and the notion of " 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 locus of any particular wave-lengths X travels, we see from (13) that U = 5C. The actual fact that when a limited group of v.aves 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. 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 for¥ard from the rear. General explanations, not restricted to the case of water-waves, have been given by Stokes, Rayleigh, and others. If the wavelength
X bo regarded as a function of a: and /, we have ax, ,, a
(14)
since X 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
—A .§ h— —^ ex
dt^'dx'Ox
^d- dx
(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−λ dc/λd (16)
If a curve be constructed with λ as abscissa and c as 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 parabola, and the intercept is half the ordinate, in accordance with the relation U = Jc, already remarked. The physical importance of the motion of group-velocity was pointed out by O. 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 a whole with the velocity c.
The preceding investigations enable us to infer the effect of a pressure-disturbance travelling over the surface of still water with, say, a constant velocity c in the direction of s:-negative. The ab-
normal pressure being supposed concentrated on an infinitely narrow band of the surface, the elevation 7; 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 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 consideration that the phase at P is " stationary " for variations in the position of Q. Now if / be the time which the source of disturbance has. taken to travel from Q to its actual position O, it appears from (12) that the phase of the waves at P,
originated at Q, is gC/^x+j-r, where a: = QP. The condition for stationary phase is therefore
x = 2x[t.
(17)
In this differentiation, O and P are to be regarded as fixed; hence x=c, and therefore 0Q = c/ = 2PQ. We have already seen that the wave-length at P is such that PQ = U/, where U is the corresponding group- velocity. Hence the Fig. 3. ​ (18) Fig. 4. — iSinS-, wave-length X at points to the right of O is uniform, being that proper to a wave- velocity c, viz. X = 27rcVg. The disturbance is therefore followed by a train of waves of approximately simple harmonic profile, of the length indicated. An approximate calculation shows that, e.xcept in the immediate neighbourhood of the source of disturbance, the surface-elevation is given by 2P0. er "=^1?='"!?. • • • where x is now measured from O, and Poi^fpdx) 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 iny. ferred by integration. The annexed figure gives a representation of a particular case, obtained by a more exact process. The pressure is here supposed 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 ij{2gh) the results are modified if the velocity c 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 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 paint 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 • • • (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 peculiar system of waves which is seen to accompany a ship. The configuration of the wave-system 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 originally given by Lord Kelvin (1887) in that it indicates the difference 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 too previous positions of the moving pressure-point which have transmitted vibrations of After V Walfrid Ekman, 0» S/aKoiwrjr stationary phase to any given Waves, n Runmns Water. p^j^^^ p ^.^^.^ ^^^ ^^^^^ ^^, j^^ riG. 5. figure. When the depth is finite the configuration is modified, and if it be less than c'/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 ships. This is accounted for by the energy of the new wave groups 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 surface-tension and the theory of " ripples " see Capillary Action. § 5. Surface-Waves of Finite Height. The foregoing results are based on the iissumption 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 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 -^e'sink{a-ct), y = b — k -^e'cosk(, a-ct),
(i) where (a, b) is the mean position of the particle, k = 2Trl; and the wave-velocity is ^ = V(g/*) = V(gX/2^). -.
(2) xxvin 8 The lines of equal pressure, among which is included of course the surface-profile, are trochoidal curves. The extreme form of waveprofiie is the cycloid, with the cusps turned upwards. The mathe-FiG. 6. matica! 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 o! 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 l-ourier series, thus y = aco5kx+ka'^cos2kx+ik^a'cosikx+ . . .,
(3) the corresponding wave-velocity being appro.ximately ^-vm+m' - ^^) where X=27r/fe. 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 blunt angle of 120°. Ac- cording to the calculations of J. H . Michell (1893), the height is then about one-seventh of the wave-length, and the wave-velocity 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 water-particles 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 obscr'ed in the case of sea-waves. 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 wave-length is supposed very great
compared with the depth (h). We have then practically the " solitary wave " to which attention was first directed by J. Pjc 8 Scott Russell (1844) from observation. The theory has been worked out by J. Boussinesq (1871) and Lord Rayleigh. The surface-elevation is given by V^asecWhMb), (5) provided 6' =/!=(/! -fa)/3o.
(6) and the velocity of propagation is c = ^lg{h+a)] (7) In the extreme form a = h and the crest forms an angle of 120°. It appears that a solitary wave of depression, of permanent type, is impossible. Bibliography. — Experimental researches: E. H . u . W . Weber, Wellenlehre (Leipzig, 1825); J. Scott Russell, " Report on Waves, " Brit. Assoc. Rep. (1844). Theoreticalv, 'orks:S. D . Poisson, " Memoire sur la theorie du son, " /. de I'ecole polyl. 7 (1807); " Mem. sur la theorie des ondes, " Mem. de I'acad. roy. des sc. I (1816); A. Cauchy. " Mel?i. sur la theorie des ondes, " Mem. de I'acad. roy. des sc. (1827); Sir G. B . Airy, " Tides and Waves, " Er.cyd. Metrop. (1845). Many classical investigations are now most conveniently accessible 2a ​in the following collections: G. Green, Math. Papers (Cambridge, 1871); H. v. Helmholtz, Gesammelte Abhandlungen (Leipzig, 1882–1895); 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 Körper, 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.)