Now in § 251 we have seen that the value of

depends on the

scale of temperature adopted, since the value of Q depends on this scale. Hence the dimensions of depend on the temperature scale. We do not, however, know the dimensions of temperature, as measured on the ordinary gas-thermometer scale, in terms of the fundamental units of length, mass, and time, and so we are reduced to using a symbol [8] for the unknown dimensions of temperature. Since the thermal unit depends on the mass of water taken, as well as on the unit of temperature, we have

The symbol here plays the part of a fourth fundamental unit, and Professor Rücker has proposed to call it a secondary fundamental unit. There is no doubt that it is only the limit of our knowledge as to the nature of temperature which prevents our expressing [0] in terms of [Z], [M], and [7]. For instance, we have in § 257 supposed that in the case of a gas the temperature is proportional to the mean kinetic energy of translation of the molecules. Hence we might measure temperatures by the mean kinetic energy of a molecule of a gas when at that temperature, and we should on this scale have

[0]=[ML2T-2].

As yet, such a method of measuring temperature is not warranted by our knowledge of the molecular conditions of gases, to say nothing of liquids and solids. It is, therefore, better to retain, when dealing with dimensional formulæ involving temperature, the symbol [0] for the dimension of the unit of temperature.

Since specific heat is the quantity of heat required to raise unit mass through a temperature of one degree,

So that specific heat has no dimension, and is therefore a mere number. This is at once evident, if we remember that specific heat may also be defined as the ratio of the heat required to raise a given mass of the substance through a given range of temperature, to the heat required to raise an equal mass of water through the same range.

Latent heat being the quantity of heat required to convert unit mass of the substance from one state to the other,

Since the coefficients of expansion are

we have

Original length (or volume) Increase in Temperature'

The quantity of heat, Q, which passes in a time, 7, through a slab of area A, thickness d and conductivity k, when the difference of temperature between the opposite faces is 0, is given by

BOOK III

WAVE-MOTION AND SOUND

PART I-WAVE-MOTION

CHAPTER I

WAVE-MOTION AND WATER WAVES

266. Wave-Motion. We have in Book I. chap. vii. considered the periodic motion of a single particle or rigid body; we have now to consider in some detail the resultant motion when the various particles of a medium are executing periodic motions, but the phase ($50) of the motions of the various particles is not the same for all, but are related to one another in certain definite ways.

Suppose we have a number of particles arranged, when at rest, at equal distances along a line AB (Fig. 214), and that these particles all

execute S.H.M.'s ($50) of equal amplitude and period along lines at right angles to AB, but in such a way that the phase of each successive particle, counting from A, differs from that of the preceding particle by a

constant amount.

Thus if the constant difference in phase is 30°, when the particle I is at its median position, the position of the others will be as shown by the dots in the figure. The displacement of particle 2 at any moment is equal to the displacement of particle 1 at 1/12 of the periodic time (7)

later, since 30° is 1/12 of 360°. Similarly, particle 3 is displaced to the

amount that particle I will be at 27/12 from the start, and so on.

Hence

the curve drawn through the positions of the particles at any instant will be a harmonic curve (§ 52). Particle 13 will at every moment be in exactly the same state as particle 1, particle 14 as particle 2, and so on; for, as their phases differ by a whole period, they will be equally displaced and moving in the same direction.

In Fig. 215 the positions of the particles are shown at successive T

12

intervals of up to half a complete vibration from the positions depicted in the first line, the direction of motion at the given instant being indicated by an arrow-head. It will be seen that the curve drawn through the particles can in each case be obtained by displacing the curve for the preceding configuration to the right, and hence, as the motion goes on, the curve connecting the particles appears to move steadily to the right. The distance through which it moves during one complete period of one of the moving particles is equal to the distance between two particles which are moving at every instant in the same direction and are equally displaced on the same side of their mean positions. This distance through which the curve, called in this case a wave, moves during a complete period of one of the moving particles is called the wave-length of the motion. The wave-length may also be defined as the distance between one particle and the next one that is displaced from its mean position to the same extent and is moving in the same direction, that is, between two consecutive particles which are in the same phase. Thus in Fig. 214 the wave-length is equal to AC or DE.

Although the form of the wave is similar to the harmonic curve, it must be remembered that the harmonic curve represents the successive displacements of a single particle, the abscissæ representing time, while the wave-form curve represents the simultaneous positions of a number of particles, the abscissæ being the distance of the mean positions of the particles measured from some fixed point. However, as all the particles move in exactly the same way, and in one whole wave-length we shall have an example of a particle in every phase of this motion, we may look upon the wave-curve as also showing us what the displacement of each particle will be at different times.

A point on the wave such as F (Fig. 214), at which the particle is at its maximum positive displacement, is called a crest, while a point such as D or E, where the displacement has its maximum negative value, is called a trough. The positions of the crests and troughs appear to travel towards the right as the motion of the particles continues.

This translatory motion of the wave is not accompanied by the translation of the particles themselves, that is, although each particle moves to and fro along its own little path, yet its mean position during a complete oscillation remains unaltered. We may, therefore, define a wave as a form or disturbance which travels through a medium, and is