# 1911 Encyclopædia Britannica/Wave

WAVE.[1] 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

 $\ddot{y}=c^2y^{\prime \prime},$ (1) where $c=\sqrt{}(P/\rho).$ (2)

The general solution of (1) 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 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

 $y==cy^{\prime},\,$ (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

 $T=\frac{1}{2} \rho \int y^2 dx, \quad V=\frac{1}{2}P \int y^{\prime 2} dx$ (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 ρ0 the undisturbed density. If p depends only on the density, we may write, for small disturbances,

 $p=p_0+ks,\,$ (11)

where s=(ρ-ρ0)ρ0, 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=p0 by Boyle's Law, and therefore c=√p0 / ρ0. 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 ss alternately compressed and rarefied. When this is allowed for we have k=γp0, where γ is the ratio of the two specific heats uf the gas, and therefore c=(yp0 / ρ0). 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.).

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 wǣg, 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."