The Wave Theory: Leonhard Euler

The wave-theory was defended by:

• Franklin[8]
• the mathematician Leonhard Euler (b, 1707, d. 1783)

Euler published ‘Nova Theoria Laucis et Colorum’ [9] while living under the patronage of Frederic the Great at Berlin.

• He insisted strongly on the resemblance between light and sound: “light is in the aether the same thing as sound in air.”

He accepted Newton’s doctrine that colour depends on wavelength, he in this memoir supposed the frequency greatest for red light, and least for violet; but a few years later[10] he adopted the opposite opinion.

The chief novelty of Euler’s writings on light is his explanation of the manner in which material bodies appear coloured when viewed by white light; and, in particular, of the way in which the colours of thin plates are produced.

He denied that such colours are due to a more copious reflexion of light of certain particular periods, and supposed that they represent vibrations generated within the body itself under the stimulus of the incident light.

A coloured surface, according to this hypothesis, contains large numbers of elastic molecules, which, when agitated, emit light of period depending only on their own structure.

The colours of thin plates Euler explained in the same way, the elastic response and free period of the plate at any place would, he conceived, depend on its thickness at that place; and in this way the dependence of the colour on the thickness was accounted for, the phenomena as a whole being analogous to well-known effects observed in experiments on sound.

An attempt to improve the corpuscular theory in another direction was made in 1752 by the Marquis de Courtivron,[11] and independently in the following year by T. Melvill.[12]

These writers suggested, as an explanation of the different refrangibility of different colours, that “the differently colour’d rays are projected with different velocities from the luminous body: the red with the greatest, violet with the least, and the intermediate colours with intermediate degrees of velocity.”

On this supposition, as its authors pointed out, the amount of aberration would be different for every different colour; and the satellites of Jupiter would change colour, from white through green to violet, through an interval of more than half a minute before their immersion into the planet’s shadow; while at emersion the contrary succession of colours should be observed, beginning with red and ending in white. The testimony of practical astronomers was soon given that such appearances are not observed; and the hypothesis was accordingly abandoned.

The fortunes of the wave-theory began to brighten at the end of the century, when a new champion arose. Thomas Young, born at Milverton in Somersetshire in 1773, and trained to the practice of medicine, began to write on optical theory in 1799.

In his first paper[13] he remarked that, according? to the corpuscular theory, the velocity of emission of a corpuscle must be the same in all cases, whether the projecting force be that of the feeble spark produced by the friction of two pebbles, or the intense heat of the sun itself—a thing almost incredible. This difficulty does not exist in the undulatory theory, since all disturbances are known to be transmitted) through an elastic fluid with the same velocity.

The reluctance which some philosophers felt to filling all space with an elastic fluid he met with an argument which strangely foreshadows the electric theory of light: “That a medium resembling in many properties that which has been denominated ether does really exist, is undeniably proved by the phenomena of electricity. The rapid transmission of the electrical shock shows that the electric medium is possessed of an elasticity as great as is necessary to be supposed for the propagation of light.

Whether the electric ether is to be considered the same with the luminons ether, if such a fluid exists, may perhaps at some future time be discovered by experiment: hitherto I have not been able to observe that the refractive power of a fluid, undergoes any change by electricity.”

Young then proceeds to show the superior power of the wave-theory to explain reflexion and refraction, In the corpuscular theory it is difficult to see why part of the light should be reflected and another part of the same beam reflected;

but in the undulatory theory there is no trouble, as is shown by analogy with the partial reflexion of sound from a cloud or denser stratum of air: “Nothing more is necessary than to suppose all refracting media to retain, by their attraction, a greater or less quantity of the luminous ether, so as to make its. density greater than that which it possesses in a vacuum, without increasing its elasticity.”. This is precisely the hypothesis adopted later by Fresnel and Green.

In 1801, Young made a discovery of the first magnitude[14] when attempting to explain Newton’s rings on the principles of the wave-theory.

Rejecting Euler’s hypothesis of induced vibrations, he assumed that the colours observed all exist in the incident light, and showed that they could be derived from it by a process which was now for the first time recognized in optical science.

The idea of this process was not altogether new, for it had been used by Newton in his theory of the tides.

“It may happen,” he wrote,[15]: “that the tide may be propagated from the ocean through different channels towards the same port, and may pass in less time through some channels than through others, in which case the same generating tide, being thus. divided into two or more succeeding one another, may produce by composition new types of tide.”

Newton applied this. principle to explain the anomalous tides at Batsha in Tonkin, which had previously been described by Halley.[16]

Young’s own illustration of the principle is evidently suggested by Newton’s.

“Suppose,” he says,[17] “a number of equal waves of water to move upon the surface of a stagnant lake, with a certain constant velocity, and to enter a narrow channel leading out of the lake; suppose then another similar cause to have excited another equal series of waves, which arrive at the same channel, with the same velocity, and at the same time with the first. Neither series of waves will destroy the other, but their effects will be combined; if they enter the channel in such a manner that the elevations of one series. coincide with those of the other, they must together produce a series of greater joint elevations; but if the elevations of one: series are so situated as to correspond to the depressions of the other, they mist exactly fill up those depressions, and the surface of the water must remain smooth. Now I maintain that similar effects take place whenever two portions of light are thus mixed; and this I call the general law of the interference of light.”

Thus, “whenever two portions of the same light arrive to the eye by different routes, either exactly or very nearly in the same direction, the light becomes most intense when the difference of the routes is any multiple of a certain length, and least intense in the intermediate state of the interfering portions; and this length is different for light of different colours.”

Young’s explanation of the colours of thin plates as seen by reflexion was, then, that the incident light gives rise to two beans which reach the eye: one of these beams has been reflected at the first surface of the plate, and the other at the second surface; and these two beams produce the colours by their interference.

One difficulty encountered in reconciling this theory with observation arose from the fact that the central spot in Newton’s rings (where the thickness of the thin film of air is zero) is black and not white, as it would be if the interfering beams were similar to each other in all respects. To account for this Young showed, by analogy with the impact of elastic bodies, that when z light is reflected at the surface of a denser medium, its phase is retarded by half an undulation: so that the interfering beams at the centre of Newton’s rings destroy each other.

The correctness of this assumption he verified by substituting essence of sassafras (whose refractive index is intermediate between those of crown and flint glass) for air in the space between the lenses; as he anticipated, the centre of the ring-system was now white.

Newton had long before observed that the rings are smaller when the medium producing them is optically more dense.

Interpreted by Young’s theory, this definitely proved that the wave-length of light is shorter in dense media, and therefore, that its velocity is less. The publication of Young’s papers occasioned a fierce attack on him in the Edinburgh Review, from the pen of Henry Brougham, afterwards Lord Chancellor of England.

Young replied in a pamphlet which sold only 1 copy: There can be no doubt that Brougham for the time being achieved his object of discrediting the wave-theory.[19]

Young now turned his attention to the fringes of shadows.

In the corpuscular explanation of these, it was supposed that the attractive forces which operate in refraction extend their influence to some distance from the surfaces of bodies, and inflect such rays as pass close by. If this were the case, the amount of inflexion should obviously depend on the strength of the attractive forces, and consequently on the refractive indices of the bodies-a proposition which had been refuted by the experiments of s’Gravesande.

The cause of diffraction effects was thus wholly unknown, until Young, in the Bakerian lecture for 1803,[20] showed that the principle of interference is concerned in their formation; for when a hair is placed in the cone of rays diverging from a luminous point, the internal fringes (i.e. those within the geometrical shadow) disappear when the light passing on one side of the hair is intercepted. His conjecture as to the origin of the interfering rays was not so fortunate; for he attributed the fringes outside the geometrical shadow to interference between the direct rays and rays reflected at the diffracting eige; and supposed the internal fringes of the shadow of a narrow object to be due to the interference of rays inflected by the two edges of the object.

The success of so many developments of the wave-theory led Young to inquire more closely into its capacity for solving the chief outstanding problem of optics—that of the behaviour of light in crystals. The beautiful construction for the extraordinary ray given by Huygens had lain neglected for a century; and the degree of accuracy with which it represented the observations was unknown.

At Young’s suggestion Wollaston[21] investigated the matter experimentally, and showed that the agreement between his own measurements and Huygeus’ rule was remarkably close. “I think,” he wrote, “the result must be admitted to be highly favourable to the Huygenian theory; and, although the existence of two refractions at the same time, in the same substance, be not well accounted for, and still less. their interchange with each other, when a ray of light is made to pass through a second piece of spar situated transversely to the first, yet the oblique refraction, when considered alone, seems nearly as well explained as any other optical phenomenon.”

Meanwhile the advocates of the corpuscular theory were not idle; and in the next few years a succession of discoveries on their part, both theoretical and experimental, seemed likely to imperil the good position to which Young had advanced the rival hypothesis.

The first of these was a dynamical explanation of the refraction of the extraordinary ray in crystals, which was published in 1808 by Laplace.[22] His method is an extension of that by which Maupertuis had accounted for the refraction of the ordinary ray, and which since Maupertuis’ day had been so developed that it was now possible to apply it to problems of all degrees of complexity.

Laplace assumes that the crystalline medium acts on the light-corpuscles of the extraordinary ray so as to modify their velocity, in a ratio which depends on the inclination of the extraordinary ray to the axis of the crystal: 50 that, in fact, the difference of the squares of the velocities of the ordinary and extraordinary rays is proportional to the square of the sine of the angle which the latter ray makes with the axis.

The principle of least action then leads to a law of refraction identical with that found by Huygens’ construction with the spheroid; just as Maupertuis’ investigation led to a law of refraction for the ordinary ray identical with that found by Huygens’ construction with the sphere.

The law of refraction for the extraordinary ray may also be deduced from Fermat’s principle of least time, provided that the velocity is taken inversely proportional to that assumed in the principle of least action; and the velocity appropriate to Fermat’s principle agrees with that found by Huygens, being, in fact, proportional to the radius of the spheroid. These results are obvious extensions of those already obtained for ordinary refraction.