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ON THE

UNDULATORY THEORY OF OPTICS.

ON UNDULATIONS GENERALLY.

PROP. 1. To explain the nature of an Undulation.

1. The characteristic of an undulation is, the continued transmission in one direction of a relative state of particles amongst each other, while the motion of each particle separately considered is a reciprocating motion. The disturbance of the particles from their state of rest, and their motion, may be in any direction whatever.

2. For example: in fig. 1, let the line (a) represent a number of particles in their position of rest: and suppose that in consequence of a disturbance they are at a given time Tin the position (3): at the time T+, in the position (y) :

at the time T+27, in the position (5): at the time T+

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in the position (e): and at the time T+7, in the position (5): and in intermediate positions at times intermediate to these. At the time T the particles are in the state of greatest condensation about a, a, and a". Suppose we fix our attention on one of these condensed groups, as for instance that of which a' is the center. At the time T+ the center of the

T

4

condensed group has glided from a to d', not by the motion of all the particles in that direction, but by such a difference of motions that the particles about a' are not so close together

as they were, and the particles about d' are closer together

2T

than they were. At the time T+ the point of greatest

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condensation has advanced to g', precisely the point where at the time T there was the least condensation: at the time 3T

T+, it has advanced to k': and at the time T+7, to a".

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The particles are now, it may be observed, in just the same state as at the time T, for a was then the center of a condensed group. After this, everything goes on in the same manner, beginning at the time T+7, as it did beginning at the time T. All that we have said with respect to the condensed mass about a' applies to those about a, a", and a"". Now if these motions were really going on before our eyes, we should see several condensations (not the condensed particles) passing uniformly and continuously from the left to the right of the line of particles.

But if we fix our attention on any one of these particles, we shall see that it has a reciprocating or oscillating motion. The particle a is advancing from T to T+7, when it has

3T

4

attained its greatest advance: it recedes then to T+ : it then advances again. The particle d advances from T (when it is at its minimum advance) to T+ it then recedes

2T

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to T+7. The particle g recedes to T+, then advances

3T

to T+ then recedes. And so for the others. The vary" 4

ing state of particles which we have here supposed, satisfies therefore the conditions mentioned in (1), and therefore this is an instance of undulation, the motion of every particle being backwards and forwards in the same line as the direction of transmission of the wave*.

* This is the kind of undulation which in the air produces sound, and is the only kind which, till within a few years, was used for the explanation of the phenomena of Optics.

3. As another example, let (B), (v), (d), (e), (5), of fig. 2, represent successive states of the particles which when at rest were in the position (a). If we fix our attention on one of the most elevated parts, as for instance k, at T, we find that

2T

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at T+the elevation has passed to a'; at T+ to d': &c.: though the particles have had no motion whatever in that direction. And if these motions were actually before us, we should see several elevations passing uniformly and continuously from the left to the right. But if we fixed our attention. on any one particle, we should see that it has an oscillating motion above and below the line. The particle a for instance, is at its greatest elevation at T+, and at its greatest de

3T 4

T

pression at T+ : d is at its greatest depression at T, at its

2T 4

greatest elevation at T+ and at its greatest depression at T+7: and so for the others. This varying state of particles is therefore another instance of undulation, the motion of every particle being at right angles to the direction of transmission of the wave.

We might conceive more complicated cases of undulation, as when the motion of the particles is compounded of the two motions supposed in these two cases*; or when there is one motion similar to that represented in fig. 2 in the plane of the paper, and another perpendicular to that planet; &c. The last of these suppositions is that to which we shall hereafter refer the phænomena of polarization, and of Optics in general.

PROP. 2. The length of a wave does not depend on the extent of vibration of each particle.

4. It is easily seen that the interval between corresponding points of two waves of condensation in fig. 1 (which is

* This is the kind of undulation which takes place on the surface of deep water in a calm.

This is the undulation of a musical string,

the distance from a to a', a to a", &c. at T, or the distance from d to d', d' to d", &c. at T+, &c.) is wholly independent of the extent of vibration of each particle. For if each particle vibrated only half as far as is now supposed, still at T, a would be a point where the particles are most condensed, and a' would be the next point where they are most condensed, and a" the next, &c. The interval between similar points of two waves (which we shall call the length of a wave, and shall always denote by the letter λ) would be the same as at present: the only difference would be that the particles about a, a', &c. would not be so closely condensed, nor those about g, g', &c. so widely separated as at present. Similarly the length of a wave in fig. 2 would be unaltered if the vibration of the particles were altered in any ratio: the only difference would be that the elevation of the high points and the depression of the low points would be altered in that ratio.

PROP. 3. The length of a wave depends on the velocity of transmission, and on the time of vibration of each particle. 5. In the cases both of fig. 1 and of fig. 2 (and in every other conceivable case of a continued sequence of waves) we see that every particle has returned to the same state at ĺ+T as at T, that is, that the vibration of every particle is completed in the time T. But in this time the wave has appeared to glide over a space equal to the interval between corresponding points of two waves, or λ. Hence we find,

Space described by the wave in the time of vibration of a particle = λ.

Velocity of wave =

λ

time of vibration of a particle'

PROP. 4. To express algebraically the transmission of an undulation.

6. The quantity for which we shall seek an expression is, the distance of any point from its point of rest, in a function of the time and of the distance of that point of rest from some fixed point. Let x be the original distance of any point in

the line (a) from some fixed point: to find an expression for its disturbance at the time t, in a function of x and t, consistent with the conditions of an undulation. By the original description of an undulation (1), putting y for the velocity of the wave's transmission, it is easily seen that whatever be the state of disturbance at the time t of a particle whose original ordinate is x, the same state of disturbance must hold at the time t+t' for a particle whose original ordinate is x + vt'. Or if express the form of the function,

$ (x, t) must = {(x+vt'), (t+t')},

whatever be the value of t'. It will be found on trial that (vt - x) satisfies this condition, being any function whatever. For putting t+t for t, and x + vt for x, it becomes

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the same as before. But it may be found analytically thus. Expanding the second side we have

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7. This expression however is too general to be of much use to us, and we will choose a particular form that will be more convenient. Suppose we fix on this condition to determine the form of the function: the vibration of each particle shall follow the same law as the vibration of a cycloidal pen

* This is the expression found, by investigation from mechanical principles, for the disturbance of the particles of air when sound passes along a tube of uniform bore, or for the disturbance of an elastic string (as that of a musical instrument) fixed at both ends.

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