By means of this apparatus Michelson has compared the length of the Metre des Archives with the wave-length of light of certain colours. He used the three coloured lights given out by cadmium vapour, and found that if AR, AG, AB are the wave-lengths of the three cadmium lines in air under standard condition, then—

with a possible error of a few tenths of a wave-length. This measurement would allow us, supposing all the copies of the metre were destroyed, to reproduce the metre with a very high degree of accuracy.

379*. Explanation of the Rectilinear Propagation of Light on the Wave Theory.-One of the chief causes why the wave theory of light was for a long time thought to be incorrect, was the difficulty of explaining why light was propagated in straight lines, and did not, as sound in general does, spread out in all directions after passing through a hole in a screen; and we are now in a position to consider this question. Let MM' (Fig. 362) be the trace of a plane wave-front at right angles to the paper, and P a point at which we require to calculate the effect

A

M

which will be produced by the wave. Now we may consider that each of the ether particles in the wave-front MM' becomes a centre of disturbance, and we then have to find what is the combined effect of all these centres on the ether at the point P. From P draw PA perpenP dicular to the wave-front, and let the distance PA be called d Next, with radii equal to d+\/2, d+2\/2, d+3\/2, &c., describe a series of circles with P as centre, cutting MM' at B, C, D, &c., and join BP, CP, DP, &c. Now since BP - AP=λ/2, the waves sent by the ether particles at A and B will reach P in opposite phases, and will therefore interfere. In the same way the waves sent from B and C will interfere, and so also the waves coming from the particles between A and B will interfere with the waves coming from the particles between B and C. Now the same argument will apply to the waves coming from all the particles on the wave-front included in a circle described about A as centre, and with radius AB on the one hand, and the particles included in the annulus or zone having radii AC and AB. Now the effect produced at P by the waves sent from all the particles in any zone will depend on two things, namely, the area of the zone, which gives the number of ether particles which are sending waves to P, and the inclination of the

FIG. 362.

line joining P to the zone to the wave-front MM'. Since this inclination increases as the zones are taken further and further from A, the magnitude of the effect produced at P by zones having equal areas will on this account gradually fall off. Since the inclination varies from one zone to the next at first quite rapidly, but, as we shall see later, this change very soon becomes excessively small, it follows that the difference between the effects produced by equal areas of consecutive zones is at first considerable, but soon becomes inappreciable as we get away from A. We have next to calculate the areas of the successive zones. PB=d+X/2. Hence

AB (d+/2) - d2

= d2+dλ+λ2 / 4 – d2
= dλ,

Now

if we neglect the term involving A2, since A is a very small quantity. Also

and so on.

Hence the area of all the zones is the same.

Now if the distance AP or dis 10 cm., and X is 5× 10 5 cm. (green light), the radii of the zones have the following values :-

This table shows very clearly how the width of the zones diminishes very quickly at first, and then more slowly, and how very narrow the

zones become even at a distance of two millimetres from the point A, which is called the pole of P.

Now the effect produced at P by any given zone depends on the area of the zone and on the inclination to the line AP of the line joining P to the zone. The table given above shows that for the zones at quite a short distance from the pole A the width of the zones is very small, and hence the angles between the lines joining two adjacent zones to P and the line AP are practically the same. Thus, except in the case of the first few zones, the effects of consecutive zones at P are exactly equal and opposite, and hence the only portion of the wave MM', which contributes to the production of the disturbance at the point P, is that portion immediately surrounding the pole A. Thus if an opaque obstacle be placed at A, so as to cut off the disturbance coming from, say, the first ten zones, there will be no disturbance produced at P, for the disturbance coming from the zones, into which the rest of the wave can be divided, will neutralise each other by interference. That is, an obstacle of about 1.5 mm. diameter, if placed at A, will completely screen P. This result, of course, amounts to the same thing as the rectilinear propagation of light, for the obstacles employed when considering this phenomenon are in general larger than that given above.

We also at once see why it is that in the case of sound-waves "shadows" are so seldom formed. Thus, taking the case of a tuningfork giving the note C of 512 vibrations per second, the wave-length in air is about 66.7 cm. Hence if the point P is at a distance of 1000 cm. from the pole A, the diameter of the tenth zone is 2√10×66.7 × 1000= 16340 cm. In other words, the diameter of an obstacle to shut off the sound would have to be more than sixteen times the distance of P from the pole, and, under these circumstances, the obliquity of the disturbance coming from the zones would be so great as to make our investigation only a very rough approximation. Thus we see that the reason we do not obtain sound-shadows is that the wave-length of the disturbance is too great compared to the size of the obstacles ordinarily used. Where the obstacle happens to be very large, sound-shadows are sometimes observed; as, for instance, an intervening hill has often protected certain buildings from the aerial concussion produced by an explosion, while other buildings at much greater distances, but not in shadow, have had their windows broken.

380*. Diffraction.-To complete the discussion of the production of shadows on the wave theory, we must now briefly consider the phenomenon observed in the immediate neighbourhood of the edge of a shadow, and also what happens when the size of the obstacle is less than the diameter of, say, ten zones, so that the disturbance is not completely cut off from the point P by the intervention of the obstacle.

We will first consider the case of a parallel beam of light which is intercepted by an opaque object, of which one edge is a straight line.

M

Let N (Fig. 363) be the section of the edge of the obstacle, NM', taken at right angles to the paper, and P the point where the illumination is to be calculated. If no obstacle were present, we might divide the incident wave MM' into half wave-length zones, just as in the previous section. Let the amplitude of the vibration which reaches P when no obstacle is present be A, so that the intensity of the illumination at P is A2 (§§ 309, 359). Now, when the obstacle is so placed as to exactly cover half the zones, that is, when the edge passes through the pole of P, the amplitude of the disturbance produced at P will be reduced to a half, and therefore the intensity of the illumination will be A/4, that is, reduced to a quarter.

N

B

A

M

FIG. 363.

Now let the obstacle be gradually moved down till the edge N coincides with B', that is, till the first zone is completely uncovered. The result will be that the illumination at P will increase and become considerably greater than A2. The reason is that the illumination is a2 when the zone B'C' is also uncovered, and this zone affects P in the opposite phase, and therefore decreases the disturbance produced by the central zone. If the edge is now lowered to c' the intensity of the illumination will gradually decrease, and reach a minimum value which is less than A2, for the next most important zone, namely C'D', is covered, and this would increase the disturbance at P if it were in action. Proceeding in this way, we see that the illumination at P will pass through a number of maxima and minima. The variation from the value A2 will, however, become less and less as more zones are uncovered, and when about ten zones are uncovered, the illumination will remain constant at the value A2.

When the edge has uncovered the first zone, the illumination at P will be the same as that at a point P1, where PP1 = AB, before the edge was moved from the position shown in the figure. Hence, since the illumination at P when the edge is at B' is greater than A2, it follows that the illumination at P1, when the edge is at the point A, must also be greater than A2. Thus if a screen be placed at PP, there will be a series of maxima and minima of illumination near the points P1, P2 P3, &C., when the edge is at A.

Next, to examine the illumination which will be produced on the portion of the screen below P, that is, within the geometrical shadow of the obstacle M'N. When the edge is at A, the illumination at P is A2/4, and as the edge is moved up the central zones are gradually covered, and hence the intensity of the disturbance sent to P gradually falls off.

The decrease in the illumination within the geometrical shadow is continuous, that is, there are no maxima and minima. The reason is that,

EDCBN

M

supposing the edge to occupy the position shown in Fig. 364, then, starting from N, we may divide the remainder of the wave-front into half-period zones NB, BC, CD, &c. Of these zones, each will produce a greater effect than the next, but adjacent ones will send to P waves in opposite phase. Thus the illumination sent to P will be practically the difference of the effects of the first two zones, or at any rate of the first three or P four. As the distance NA is increased, that is, as Pis taken further and further inside the geometrical shadow, the difference between the effect produced by the first two zones will decrease, just as in § 379 we found that consecutive zones, after about the tenth from the pole, had equal and opposite effects.

A

M'

FIG. 364.

Thus the wave theory indicates that the shadow cast by a sharp edge when illuminated by parallel light, or, what comes to the same thing, light from a point, source, or narrow slit at a considerable distance, is not quite sharp. Outside the geometrical shadow will be a series of light and dark bands, and inside the light will not cease suddenly, but will fall off rapidly.

The intensity of the illumination on a screen placed at a distance of one metre from a diffracting edge, and illuminated by a parallel beam of light, is shown by means of a curve in Fig. 365. It will be seen that the

illumination at the edge of the geometrical shadow is a quarter of the illumination at some distance from the edge, that is, of the illumination which would occur if the diffracting obstacle were removed.