due to the parts of the medium performing in succession certain periodic motions about their mean positions.

In the case of wave motion considered above, the particles all vibrated at right angles to the direction in which the wave moves, and this form of wave-motion is said to be due to transverse vibrations. If the motion of each particle takes place in the direction in which the wave moves, then the vibration is said to be longitudinal.

At AB (Fig. 216) the undisplaced positions of the particles are shown. If each particle now executes a S.H.M. in the direction AB, the period and amplitude being the same for all, but the phase of each particle being 30° behind that of the preceding particle, then when particle 1 is

passing through its mean position and moving towards the right, CD will represent the positions of the other particles. The positions of the particles are also shown at successive intervals of 1/12 of the period of the S.H.M. of each particle for half a complete period.

In this form of wave-motion the distances between adjacent particles alter, so that the particles are alternately crowded together and spread out. A point where at any instant the crowding together is a maximum is called a condensation, while a point where the distance between adjacent particles is a maximum is called a rarefaction. These play the same parts in longitudinal wave-motion as do the crests and troughs in transverse wave-motion.

The definition of wave-length, given with reference to transverse vibrations, applies also to longitudinal vibrations. The most convenient manner of studying longitudinal vibrations is to employ a curve of which the ordinates indicate the displacements from their mean positions of the different particles at any time. Such a curve is obtained if, at the mean or undisturbed position of each particle, we erect a perpendicular in the positive or negative direction according as the displacement of the

particle is in the positive or negative direction, and having a height equal to the displacement of the particle from its mean position. The curve obtained by joining the extremities of these ordinates is shown at EFG (Fig. 216), the corresponding positions of the disturbed and undisturbed particles being shown at HK and AB. This curve is a harmonic curve, and the points where the curve cuts the axis correspond to the places where the particles are most crowded together, or most spread out. For at F the particle 7 is at its mean position, while the particle 6 is, since the corresponding ordinate ab of the curve is positive, displaced to the right by an amount equal to this ordinate, and the particle 8 is to the left of its median position by an amount equal to the ordinate cd of the curve; so that the particles are here crowded together. In the same way the particles at E and G are separated to a maximum extent. Hence F corresponds to a condensation, while E and G correspond to rarefactions. The distance between two adjacent rarefactions, such as E and G, or between two condensations, is equal to the wave-length of the wavemotion, while the distance between a rarefaction and the adjacent condensation is equal to half a wave-length.

267. Velocity of Propagation of a Wave-Frequency.—The speed at which the crest or trough in the case of a transverse wave, or the condensation or rarefaction in a longitudinal wave, moves through the medium is called the velocity of propagation of the wave-motion.

While particle 4 in Figs. 214 and 215 is making a complete oscillation, the trough of the wave will travel to the right to particle 16, that is, through a distance equal to the wave-length, X. In the same way, while particle 1 (Fig. 216) is making a complete oscillation, the condensation will travel from C to D, that is, through a distance equal to the wave-length.

Hence if 7 is the time each particle takes to complete one oscillation, in this time the wave will move through a distance equal to the wavelength. Thus if is the velocity of propagation of the wave, we v=X/7.

have

Each time that particle 10 (Fig. 214) reaches its maximum positive elongation, a crest will be passing at F, so that the interval between the passage of two successive crests is 7. Thus if n is the number of crests which pass F in a second, we have n=1/T. The same remark applies to any other particle, whether the motion is transverse or longitudinal, and the quantity n is called the frequency of the waves. Thus

The velocity with which a group of waves moves into an undisturbed portion of the medium is not necessarily equal to the velocity of the individual waves. Thus in the case of gravitational waves on a liquid, the individual waves travel twice as fast as does the front of the disturbance. Thus if we watch a short train of waves moving into

still water, the waves will appear to move through the group, dying out in front, and fresh waves appearing in the rear of the group. It can be shown that whenever the velocity of the waves varies with the wavelength, the group velocity is different from the wave velocity.

The study of waves being of very great importance in physics-for, as we shall see, sound, light, radiant heat, and many electro-magnetic phenomena are propagated by wave-motions-it will be advisable to spend some time considering this form of motion. It will add to the interest, and also to the clearness, of the study of a wave-motion if we illustrate the various points by reference to some particular form of wave-motion. Now the waves which constitute sound, light, and heat are invisible, and so for the purposes of illustration it will be better to consider the waves which may be produced at the surface of a liquid, for such waves may, with suitable arrangements, be seen by the eye.

268. Waves on the Surface of a Liquid.—In order that a wave may be formed, it is necessary that the successive particles which constitute the medium in which the wave is propagated should each in succession go through a periodic motion. Now when considering the motion of a pendulum (§ 112), we showed that the reason it executes its periodic motion is that, when the bob is displaced, a force acts on the bob tending to bring it back to its position of rest. Hence when dealing with the production of a wave-motion in a medium, we must consider how the force of restitution on the particles of the medium, which is necessary for the production of the periodic motion of these particles, is brought about.

Let

E

form DEF, or scooped out into a hollow FGH, so that the liquid particles are displaced from their positions of rest. Then, owing to the action of gravity, the particles in the portion DEF of the liquid will move back towards the level surface AB, while the particles which have been forced down owing to the production of the hollow FGH will move up. Thus when the particles of a liquid are moved, so that a portion of the surface is displaced either above or below the level of the general surface, owing to gravity a force will act tending to bring the surface back to its undisturbed position. We have therefore the conditions suitable for the production of waves on the surface of the liquid, and the existence of these waves being due to the action of gravity, they are called gravitational waves. The large waves seen on the surface of the sea are well-known examples of gravitational waves. Gravity, as was, however, first pointed out by Lord Kelvin, is not the only cause tending to bring the surface of the liquid

back to its undisturbed position. There is a second cause acting, namely, the surface tension (§ 157) of the surface film of the liquid. This surface tension acts as if there were a thin elastic membrane stretched over the surface, and it is evident that the effect of such a stretched membrane will be to tend to flatten down the portion DEF of the disturbed surface of the liquid, and to level up the portion FGH, so that in the surface tension we have also a force of restitution acting on the displaced liquid particles We have seen in § 158 that the pressure

due to the surface tension increases with increase of the curvature of the surface, so that, since the magnitude of the surface tension is small, it is only when we are dealing with waves in which the curvature is very great that we need take account of surface tension.

It can be shown,' although to do so would lead us beyond the scope of this book, that if v is the velocity of a wave along the surface of a liquid of which the depth is not less than the wave-length λ, and p and 7 are the density and surface tension of the liquid, then

ελ 2пт
+
2π λρ

From this expression it follows at once that if the wave-length is great, the fraction 27λp is small compared to gλ/2, and hence may be neglected. The fact that A is great shows that the curvature of the surface must be small, so that this result is what we should expect. On the other hand, if λ is small, then 27Ap is great compared to gλ/2′′, so that in this case surface tension plays the important part in the propagation of the waves. Such waves, in which the greater part of the force of restitution is due to surface tension, are called capillary waves or ripples.

For waves of wave-length greater than about 4 inches or 10 cm. the term 27λp may be neglected, while for waves of wave-length less than 0.1 inch or 3 mm. the term gλ/2′′ may be neglected. For waves having wave-lengths between these two limits, we have to take into account both the effect of gravity and of surface tension.

Since the velocity due to gravity alone increases as λ increases, and that due to surface tension alone increases as λ decreases, it follows that there must be a certain wave-length for which the velocity is a minimum. For wave-lengths less than this critical value the surface tension has the predominating influence, and therefore the velocity increases as λ decreases, as shown by the left-hand branch of the curve (Fig. 218), which

1 The case where gravity alone is supposed to act is considered in § 277.

2 In order to show a considerable range, instead of taking equal lengths along the axes to correspond with equal increments in the wave-length and the velocity respectively, in the figure equal lengths along the axes correspond to equal increments in the logarithms of these quantities. The scales have, however, been numbered so that we read off the wave-length and velocity direct. For an account of this method of plotting curves, a paper by Prof. C. V. Boys, in Nature for July 18, 1895, may be consulted.

For

gives the velocity of waves of different wave-lengths in water. wave-lengths greater than the critical value, gravity plays the important part, and the velocity increases as à increases, as shown by the right

hand branch of the curve. For water the minimum velocity is 23 cm. per second, or 9 inches per second.

269. Gravitational Waves.-In the case of waves for which the wave-length is so great that we may neglect the effect of surface tension, we have

2π

It will be observed that the density of the liquid is not involved in the expression for the velocity. The reason for this is the same as that which explains why it is that the period of a pendulum is independent of the mass of the bob, namely, that although the mass of the liquid to be moved is proportional to the density, yet, since the force of restitution is also proportional to the density, for it is the weight of the raised portion of the liquid, the ratio of the force of restitution to the mass to be moved is the same for all liquids, and therefore the velocity of the waves is the

same.

If the depth of the liquid is considerably less than the wave-length, the velocity is less than that given above, and is given by

v=gd,

where d is the depth of the liquid. One effect of this decreased velocity in shallow water is to make the waves in the neighbourhood of a shelving beach always move in a direction perpendicular to the shore, although at some distance out to sea they may be moving in quite a different direction. The reason is that when a wave which is moving in a direction inclined to the shore-line reaches shallow water, the end of the wave which first reaches the shallow moves more slowly than the parts which are still moving in deep water. Thus the wave gradually wheels round till it becomes nearly parallel to the shore.

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