otherwise, then from the time when demand of payment shall have been made in writing, so as such demand shall give notice to the debtor that interest will be claimed from the date of such demand until the term of payment: provided that interest shall be payable in all cases in which it is now payable by law.” Compound interest requires to be supported by positive proof that it was agreed to by the parties; an established practice to account in this manner will be evidence of such an agreement. When interest is awarded by a court it is generally at the rate of 4%; under special circumstances 5% has been allowed.
INTERFERENCE OF LIGHT. § 1. This term[1] and the ideas
underlying it were introduced into optics by Thomas Young.
His Bakerian lecture on “The Theory of Light and Colours”
(Phil. Trans., 1801) formulated the following hypotheses and
propositions, and thereby laid the foundations of the wave
theory:—
Hypotheses.
(i.) A luminiferous aether pervades the universe, rare and elastic in a high degree.
(ii.) Undulations are excited in this aether whenever a body becomes luminous.
(iii.) The sensation of different colours depends on the different frequency of vibrations excited by the light in the retina.
(iv.) All material bodies have an attraction for the aethereal medium, by means of which it is accumulated in their substance, and for a small distance around them, in a state of greater density but not of greater elasticity.
Propositions.
(i.) All impulses are propagated in a homogeneous elastic medium with an equable velocity.
(ii.) An undulation conceived to originate from the vibration of a single particle must expand through a homogeneous medium in a spherical form, but with different quantities of motion in different parts.
(iii.) A portion of a spherical undulation, admitted through an aperture into a quiescent medium, will proceed to be further propagated rectilinearly in concentric superfices, terminated laterally by weak and irregular portions of newly diverging undulations.
(iv.) When an undulation arrives at a surface which is the limit of mediums of different densities, a partial reflection takes place, proportionate in force to the difference of the densities.
(v.) When an undulation is transmitted through a surface terminating different mediums, it proceeds in such a direction that the sines of the angles of incidence and refraction are in the constant ratio of the velocity of propagation in the two mediums.
(vi.) When an undulation falls on the surface of a rarer medium, so obliquely that it cannot be regularly refracted, it is totally reflected at an angle equal to that of its incidence.
(vii.) If equidistant undulations be supposed to pass through a medium, of which the parts are susceptible of permanent vibrations somewhat slower than the undulations, their velocity will be somewhat lessened by this vibratory tendency; and, in the same medium, the more, as the undulations are more frequent.
(viii.) When two undulations, from different origins, coincide either perfectly or very nearly in direction, their joint effect is a combination of the motions belonging to each.
(ix.) Radiant light consists in undulations of the luminiferous aether.
In the Philosophical Transactions for 1802, Young refers to his discovery of “a simple and general law.” The law is that “wherever two portions of the same light arrive at the eye by different routes, either exactly or very nearly in the same direction, the light becomes most intense where the difference of the routes is a 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.”
This appears to be the first use of the word interfering or interference as applied to light. When two portions of light by their co-operation cause darkness, there is certainly “interference” in the popular sense; but from a mechanical or mathematical point of view, the superposition contemplated in proposition viii. would more naturally be regarded as taking place without interference. Young applied his principle to the explanation of colours of striated surfaces (gratings), to the colours of thin plates, and to an experiment which we shall discuss later in the improved form given to it by Fresnel, where a screen is illuminated simultaneously by light proceeding from two similar sources. As a preliminary to these explanations we require an analytical expression for waves of simple type, and an examination of the effects of compounding them.
§ 2. Plane Waves of Simple Type.—Whatever may be the character of the medium and of its vibration, the analytical expression for an infinite train of plane waves is
| A cos 2πλ(Vt − x) + α | (1), |
in which λ represents the wave-length, and V the corresponding velocity of propagation. The coefficient A is called the amplitude, and its nature depends upon the medium and may here be left an open question. The phase of the wave at a given time and place is represented by α. The expression retains the same value whatever integral number of wave-lengths be added to or subtracted from x. It is also periodic with respect to t, and the period is
| τ = λ/V | (2). |
In experimenting upon sound we are able to determine independently τ, λ, and V; but on account of its smallness the periodic time of luminous vibrations eludes altogether our means of observation, and is only known indirectly from λ and V by means of (2).
There is nothing arbitrary in the use of a circular function to represent the waves. As a general rule this is the only kind of wave which can be propagated without a change of form; and, even in the exceptional cases where the velocity is independent of wave-length, no generality is really lost by this procedure, because in accordance with Fourier’s theorem any kind of periodic wave may be regarded as compounded of a series of such as (1), with wave-lengths in harmonical progression.
A well-known characteristic of waves of type (1) is that any number of trains of various amplitudes and phases, but of the same wave-length, are equivalent to a single train of the same type. Thus
| ΣA cos | 2π | (Vt − x) + α = ΣA cos α.cos | 2π | (Vt − x) − ΣA sin α.sin | 2π | (Vt − x) |
| λ | λ | λ |
| = P cos 2πλ(Vt − x) + φ | (3), |
where
| P2 = (ΣA cos α)2 = Σ(A sin α)2 | (4), |
| tan φ =Σ(A sin α)Σ(A cos α) | (5). |
An important particular case is that of two component trains only.
| A cos | 2π | (Vt − x) + α + A′ cos | 2π | (Vt − x) + α′ |
| λ | λ |
| = P cos | 2π | (Vt − x) + φ , |
| λ |
where
| P2 = A2 + A′2 + 2AA′ cos (α − α′) | (6). |
The composition of vibrations of the same period is precisely analogous, as was pointed out by Fresnel, to the composition of forces, or indeed of any other two-dimensional vector quantities. The magnitude of the force corresponds to the amplitude of the vibration, and the inclination of the force corresponds to the phase. A group of forces, of equal intensity, represented by lines drawn from the centre to the angular points of a regular polygon, constitute a system in equilibrium. Consequently, a system of vibrations of equal amplitude and of phases symmetrically distributed round the period has a zero resultant.
According to the phase-relation, determined by (α − α′), the amplitude of the resultant may vary from (A − A′) to (A + A′). If A′ and A are equal, the minimum resultant is zero, showing that two equal trains of waves may neutralize one another. This happens when the phases are opposite, or differ by half a (complete) period, and the effect is that described by Young as “interference.”
§ 3. Intensity.—The intensity of light of given wave-length must depend upon the amplitude, but the precise nature of the relation is not at once apparent. We are not able to appreciate by simple inspection the relative intensities of two unequal lights; and, when we say, for example, that one candle is twice as bright as another, we mean that two of the latter burning independently would give us the same light as one of the former. This may be regarded as the definition; and then experiment may be appealed to to prove that the intensity of light from a given source varies inversely as the square of the distance. But our conviction of the truth of the law is perhaps founded quite as much upon the idea that something not liable to loss is radiated outwards, and is distributed in succession over the surfaces of spheres concentric with the source, whose areas are as the squares of the radii. The something can only be energy; and thus we are led to regard the rate at which energy is propagated across a given area parallel to the waves as the measure of intensity; and this is proportional, not to the first power, but to the square of the amplitude.
§ 4. Resultant of a Large Number of Vibrations of Arbitrary Phase.—We have seen that the resultant of two vibrations of equal amplitude
- ↑ The word “interference” as formed, on the false analogy of such words as “difference,” from “to interfere,” which originally was applied to a horse striking (Lat. ferire) one foot or leg against the other.
