Undulation

“ A breaking wave concentrates its transmitted force on a portion of the water forming the undulation, which, consequently, strikes a more powerful blow over a limited area against any structure than the more distributed shock of a simple undulation beating against a vertical wall. Moreover, the recoil of broken waves down a sloping beach or rubble mound produces a greater scour than the simple reflection of an undulation from a vertical wall, especially where the depth is sufficient to provide a cushion of water below the undulation, protecting the toe of the wall from the wash of recoil. ”

“ The duration of each undulation, as we have seen, does not depend on the greater or less velocity with which the [p446] agitation is propagated along the fluid, but merely on the duration of the previous oscillation which gave it birth ”

“ According to the theory of undulation, there is a material medium which fills the space between the two bodies, and it is by the action of contiguous parts of this medium that the energy is passed on, from one portion to the next, till it reaches the illuminated body. ”

“ Newton and subsequent physicists558 have measured light undulations, and by several methods we learn the velocity with which light travels. Since an undulation of the middle green is about five ten-millionths of a metre in length, and travels at the rate of nearly 300,000,000 of metres per second, it follows that about 600,000,000,000,000 undulations must strike in one second the retina of an eye which perceives such light. ”

“ By an experiment proposed by Arago, and executed with consummate skill by Foucault and Fizeau, this question was brought to a crucial test, and decided in favour of the theory of undulation.
In the present instance also the two theories are at variance. Newton assumed that the action which produces the alternate bright and dark rings took place at a single surface; that is, the second surface of the film. The undulatory theory affirms that the rings are caused by the interference of waves reflected from both surfaces. This also has been demonstrated by experiment. ”

“ In order to obtain a correct idea of the manner in which the oscillation of a small solid body occasions undulations in an elastic fluid, it has been only necessary to consider a complete oscillation of the solid plane, which produces an entire undulation. If we suppose the oscillations of the plane to be continually repeated, we shall have a series of undulations instead of a single one: and they will follow each other without intermission, provided that the vibrations of the particle first agitated have been regular. ”

“ In the undulatory theory, on the other hand, all tends to unity and simplicity. We explain reflection and refraction by undulations; when we come to thin plates, the requisite “fits” are already involved in our fundamental hypothesis, for they are the length of an undulation; the phenomena of diffraction also require such intervals; and the intervals thus required agree exactly with the others in magnitude, 108 so that no new property is needed. Polarization for a moment checks us; but not long; for the direction of our vibrations is hitherto arbitrary;—we allow polarization to decide it. ”

“ Going back to our illustration of the eddies on a pool of water, it is easy to prove that by throwing a second stone into the water we form another series of undulations; which are mutually destroyed when they encounter each other. It is the same with the 83peculiar fluid which, existing throughout space, is thrown in a state of undulation by incandescent bodies; by opposing one set of waves to another we obtain rest as a result. ”

“ Now, observe, if the precipices which we have just been describing had been broken when their substance was in a hard state, there appears no reason why any connexion should be apparent between the energy of undulation and these broken rocks. ”

“ Since, by the hypothesis, they differ half an undulation in their progress, the semiundulations of the one, which tend to cause in the particles of ether a motion in one direction, coincide with the semiundulations of the other, which urge them in the opposite direction, and these two forces hold each other in equilibrium, so that the motion is wholly destroyed in the whole extent of these two systems of waves, except the two extreme semiundulations, which escape from the interference. ”

“ But when the air is crossed in every direction by small currents of hotter air, the sonorous undulation is divided into two undulations where the density of the medium changes abruptly; partial echoes are formed that weaken the sound, because one of the streams comes back upon itself ”

“ The undulations which, coming from a distant centre, fall upon such a medium as we have described, are, it appears from the principles of mechanics, propagated in a manner quite different from anything which had been anticipated. The “surface of the waves” (that is, the surface which would bound undulations diverging from a point) , is a very complex, yet symmetrical curve surface; which, in the case of uniaxal crystals, resolves itself into a sphere and a spheroid ”

“ In the next place, the hypothesis of undulations similarly fails. First, because the motion assumed is not a vera causa. If it be answered that an all-pervading aether and its system of undulations are otherwise established, by the large array of well-observed phenomena shown to be in exact accordance both in quantity and in quality with the mathematical deductions from such postulated undulations, the rejoinder is, that such undulations as have thus been established are demonstrably incapable of inducing anything in the slightest degree analogous to gravitation ”

“ And this erroneous notion is, indeed, the current conception.
If a stone falls into the water, it is the cause of the undulations, but not their creator. It is only a co-operator, for the liquid and elastic qualities of the water also act as a cause. If the stone falls into butter, it creates at best but one undulation, and if this stony creator falls on the hard ground, it is all up with the creation of undulations. This shows that causes are not creators, but rather effects which are not effected, but effect themselves. ”

“ The undulations are supposed to take place, not in the direction of propagation, as in the air waves constituting sound, but transversely, and the various phenomena of refraction, polarization, interference, etc., are attributable to the different affections of these undulations in different circumstances of propagation. ”

“ No one supposes the agreement of the phenomena of light with the theory of undulations to be merely fortuitous. It must arise from the actual identity of some of the laws of undulations with some of those of light; and if there be that identity, it is reasonable to suppose that its consequences would not end with the phenomena which first suggested the identification, nor be even confined to such phenomena as were known at the time. ”

“ They are not substances, but agencies. These Imponderable Agents are not properly called Imponderable Fluids. This I conceive that I have proved.” Nothing can be more philosophical. But if the luminiferous ether is not matter, and fluid matter, too, what is the meaning of its undulations? Can an agency undulate? Can there be alternate motion forward and backward of the particles of an agency? And does not the whole mathematical theory of the undulations imply them to be material? Is it not a series of deductions from the known properties of elastic fluids? ”

“ What is true of the undulations of an India-rubber tube applies to all undulations whatsoever. Water-waves, for example, obey the same laws, and the coalescence of direct and reflected waves exhibits similar phenomena. This long and narrow vessel with glass sides, Fig. 40, is a copy of the wave-canal of the brothers Weber. It is filled to the level A B with colored water. By tilting the137 end A suddenly, a wave is generated, which moves on to B, and is there reflected. By sending forth fresh waves at the proper intervals, the surface is divided into two stationary undulations. ”

“ These describe undulations on the cylinder, and the undulations correspond to equal periods of time. No instruments vibrate with greater uniformity than tuning-forks. Duhamel was the first to apply to one of the limbs of a tuning-fork a small marker, and to bring this marker against a rapidly moving surface, like the surface of this blackened cylinder. Undulations or waves are thus described. The more rapid the movement of the cylinder, the longer will be the waves, as you see in the experiment, and as are represented in this diagram (Fig. 26) . ”

“ It is thus that sympathetic undulations are excited by a body vibrating near insulated tended strings, capable of following its undulations, either by vibrating entire, or by separating themselves into their harmonic divisions. If two chords equally stretched, of which one is twice or three times longer than the other, be placed side by side, and if the shorter be sounded, its vibrations will be communicated by the air to the other, which will be thrown into such a state of vibration that it will be spontaneously divided into segments equal in length to the shorter string. ”

“ The velocity of transverse undulations in our hypothetical medium, calculated from the electro-magnetic experiments of MM. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena. ”

“ And then, hark, 'twas Undine's voice which rang through the uproar; 'twas Undine herself who, as the moon swam clear of the clouds, was seen standing on the heights above the valley. 'Twas she, in sooth, who rebuked and threatened the floods below; and the menacing column of water vanished, murmuring and muttering, and the streams flowed gently away in the moonlight. Like a white dove Undine flew down from the height; she laid hold on the knight and Bertalda, and bore them with her to a green and grassy spot on the hill. There she refreshed their weariness and dispelled their fears ”

“ Yet the doctrine that this is so, as well as the law of the force, were established only by this and the like demonstrations. The doctrine which is the hypothesis of the deductive reasoning, is the inference of the inductive process. The special facts which are the basis of the inductive inference, are the conclusion of the train of deduction. And in this manner the deduction establishes the induction. The principle which we gather from the facts is true, because the facts can be derived from it by rigorous demonstration. Induction moves upwards, and deduction downwards, on the same stair. ”

“ To this power of propagation he gave the name 'specific inductive capacity.' He then glanced at conduction in its relation to induction, and generalised thus: 'Can we not, by a gradual chain of association, carry up the discharge from its occurrence in air through spermaceti and water to solutions, and then on to chlorides, oxides, and metals, without any essential change in its character?' The action of the particles of the best conductor differs, according to Faraday, only in degree from that of the particles of the insulator. ”

“ It was these very monstrous forms which urged him to cross the flood, as he bethought him that Undine might be among them, alone in her agony.
Now, as he grasped the stout branch of a pine and stood, supporting himself by it in the midst of the current, which only with all his force could he withstand; and while yet with unblenching courage he pressed further into the stream: he heard a soft voice which said to him, "Venture not, venture not; full crafty is that old man, the stream!" ”

“ To substitute real examples for symbolical ones, the following are three uniformities, or call them laws of nature: the law that air has weight, the law that pressure on a fluid is propagated equally in all directions, and the law that pressure in one direction, not opposed by equal pressure in the contrary direction, produces motion, which does not cease until equilibrium is restored. From these three uniformities we should be able to predict another uniformity, namely, the rise of the mercury in the Torricellian tube. ”

“ Thus we have the conclusion that by the process of inversion we get from a distribution in equilibrium, on a conductor of any form, an induced distribution on the inverse surface supposed insulated and conducting; and conversely we obtain from a given induced distribution on an insulated conducting surface, a natural equilibrium distribution on the inverse surface. In each case the inducing charge is situated at the centre of inversion. ”

“ True, he states the method or law of that change. But the method is discovered by induction. Induction is in turn annihilated by evolution. The fabric as it rises loses its foundation, and floats in the air, a baseless vision. ”

“ There is, however, between Colligation and Induction, a real correlation, which it is important to conceive correctly. Colligation is not always induction; but induction is always colligation. The assertion that the planets move in ellipses, was but a mode of representing observed facts; it was but a colligation; while the assertion that they are drawn, or tend, [pg 312] towards the sun, was the statement of a new fact, inferred by induction. But the induction, once made, accomplishes the purposes of colligation likewise. ”

“ As the power of induction is proportional to the tension of the electricity, and as the tension is continually diminished by the resistance of the wires, the induction is continually diminished and requires a longer time. ”