that the velocity of the vibrating particle at each instant is
proportional to the sine of an arc, representing the time elapsed
from the beginning of the motion, taking the circumference for the
whole time required for the return of the particle to the same point,
that is, the time occupied by two oscillations, the one forwards
and the other backwards. Such is the law according to which I have
calculated the formulas which serve to determine the effect of any
number of systems of waves of which the intensities and the relative
positions are given. These formulas will be found in the Annals of
Chemistry, vol. xi., page 254: [they may be applied with security to
the phenomena there considered, though the perfect accuracy of the
hypothesis in all possible cases may be questioned, upon the grounds
of the microscopical observations on the motions of vibrating chords,
published by Dr. Young in the Philosophical Transactions for 1800.
TR.] Without entering into the details of the calculations, I think
it necessary to show in what manner the nature of the undulation
depends on the kind of motion of the vibrating particles.
Let us suppose, in the fluid, a little solid plane which is removed
from its primitive position, towards which it is urged by a force
proportional to the distance. At the beginning of its motion, the
accelerative force produces in it an infinitely small velocity only;
but its action continuing, the effects become accumulated, and the
velocity of the solid plane goes on continually to increase, until
the moment of its arrival at [p117] the position of equilibrium, in
which it would remain, but for the velocity which it has acquired;
and it is by this velocity only, that it is carried beyond the point
of equilibrium. The same force which tends towards this point,
and which now begins to act in a contrary direction, continually
diminishes the velocity, until it is completely annihilated; and then
the force continuing its action produces a velocity in the contrary
direction, which brings the plane back to its place of equilibrium.
This velocity again is very small at the commencement of the return
of the particle, or plane, and increases by the same degrees as
it had before diminished, until the instant of the arrival of the
particle at the neutral point, which it passes with the velocity
previously acquired: but when it has passed this point, the motion is
diminished more and more by the effect of the force tending towards
it, and its velocity is reduced to nothing when it arrives at the
place of the commencement of the motion. It then recommences, at
similar periods, the series of motions which have been described,
and would continue to oscillate for ever, but for the effect of
the resistance of the surrounding fluid, the inertia of which
continually diminishes the amplitude of its oscillations, and finally
extinguishes them at the end of a longer or shorter time, according
to circumstances. [It must not be inferred from this explanation,
that the particles of a fluid transmitting an undulation have any
tendency to vibrate for ever: on the contrary it has been admitted by
the best writers on the theory of sound, that all the motions which
constitute it, as considered in a fluid, are completely transitory
in their nature, and have no disposition to be repeated after having
been once transmitted to a remoter part of the fluid. TR.]
Let us now consider in what manner the fluid is agitated by these
oscillations of the solid plane. The stratum immediately in contact
with it, being urged by the plane, receives from it at each instant
the velocity of its motion, and communicates it to the neighbouring
stratum, which it forces forwards in its turn, and from which the
motion is communicated successively to the other strata of the
fluid; but this transmission of the motion is not instantaneous,
and it is only at the end of a certain time that it arrives at a
determinate [p118] distance from the centre of agitation. This
time is the shorter, as the fluid is less dense, and more elastic;
that is, composed of particles which possess a greater repulsive
force. This being granted, let us assume, in order to facilitate the
explanation, the moment when the moveable plane is returned to the
initial situation, after having performed two complete oscillations
in opposite directions: at this moment, the nascent velocity, which
it had at first, is transmitted to a stratum of the fluid removed
from the centre of agitation by a distance which we may represent
by _d_. Immediately afterwards, the velocity of the moveable plane,
which has a little augmented, has been communicated to the stratum in
contact with it: “hence _it_ has passed successively through all the
following strata;” and at the moment when the first agitation arrives
at the stratum of which the distance is _d_, the second has arrived
at the stratum immediately before it. Continuing thus to divide, in
our imagination, the duration of the two oscillations of the moveable
plane into an infinity of small intervals of time, and the fluid
comprehended in the length _d_, into an equal number of infinitely
thin strata, it is easy to perceive, by the same reasoning, that
the different velocities of the moveable plane, at each of these
instants, are now distributed among the corresponding strata; and
that thus, for example, the velocity which the plane possessed at the
middle of the first oscillations in the direction of the motion, must
have arrived, at the instant in question, at the distance 3/4 _d_: so
that it is the stratum at this distance which possesses at the moment
the greatest direct velocity; and in the same manner when the plane
arrived at the limit of its first direct oscillation, its velocity
was extinguished, and the same absence of motion will be found at the
distance 1/2 _d_.
It is always supposed, that the oscillations of the plane are so
minute in comparison with the length _d_, that their extent may
be neglected in this calculation: and this hypothesis is actually
consistent with the fact, since there is every reason to suppose
that the excursions of the incandescent particles are very small
in comparison with the extent of an undulation, which, though an
extremely minute space, is still an appreciable quantity, and may be
actually measured. Besides, [p119] even if the amplitude of these
oscillations were not in the first instance so wholly inconsiderable,
it would be sufficient to consider an undulation at a greater
distance from the centre of agitation, in order that their extent
might be diminished in any required proportion.
In the second, or retrograde oscillation, the plane, returning
through the same space, must communicate to the stratum of fluid
in contact with it, and to the rest in succession, a motion in
a direction contrary to that of the first oscillation; for when
the plane recedes, the stratum in contact with it, urged against
the plane by the elasticity or the expansive force of the fluid,
necessarily follows it, and fills up the vacuum which its retrograde
motion tends to produce. For the same reason, the second stratum is
urged against the first, the third against the second, and so forth.
It is thus that the retrograde motion is communicated, step by step,
to the most distant strata: its propagation is effected according to
the same law that governs the direct motion; the only difference is
in the direction of the motions, or, in the language of mathematics,
in the sign of the velocities which are imparted to the molecules
of the fluid. We see then that the different velocities which have
existed in the solid plane, during its second oscillation, must exist
at the moment which we are considering, in the different strata
comprehended in the other half of _d_, but with contrary signs. Thus
the velocity, for example, which the plane had in the middle of the
second oscillation, which is its maximum of retrograde velocity, must
now be found in the fluid stratum situated at the distance 1/4 _d_
from the centre of agitation, while the maximum of direct velocity is
found, at the same instant, in the stratum which is at the distance
3/4 _d_ from the centre of agitation.
The extent of the fluid, agitated by the two opposite
oscillations of the solid plane, is what we call the breadth of
an _entire undulation_, and we may consequently give the name of
_semiundulation_ to each of the parts actuated by the opposite
undulations; the whole constituting a _complete oscillation_, since
it comprehends the return of the vibrating plane to the initial
situation. It is obvious, that the two semiundulations, which compose
the complete undulation, exhibit, in [p120] the fluid strata which
they contain, velocities absolutely equal in magnitude, but with
contrary signs, that is to say, carrying the particles of the fluid
in opposite directions. These velocities are the greatest in the
middle of each of the semiundulations, and decrease gradually towards
their extremities, where they entirely vanish: so that the points
of rest, and of the greatest velocities positive and negative, are
separated from each other by intervals of one fourth of an undulation.
The length of an undulation, _d_, depends on two things: first, on
the promptitude with which the motion is propagated in the fluid;
and secondly, the duration of the complete oscillation of the
vibrating plane; for the longer this duration, and the more rapid the
propagation of the motion, the greater will be the distance to which
the first agitation has been extended at the instant of the return of
the solid plane to its initial situation. If the oscillations are all
performed in the same medium, the velocity of propagation remaining
the same, the length of the undulations will be simply proportional
to the duration of the oscillations of the vibrating particles from
which they originate. As long as the vibrating particles continue
to be subjected to the same forces, it follows from the principles
of mechanics that each of their minute oscillations will occupy the
same time, whatever their extent may be; so that the corresponding
undulations of the fluid will continue to be of the same length; they
will only differ from each other in the greater or less extent of the
elementary vibrations of the particles, which will be proportional
to the extent of the luminous particles; for it appears from what
has already been stated, that each stratum of the fluid repeats
exactly all the motions of the vibrating particle. The greater
or less amplitude of the oscillations of the strata of the fluid
determines the degree of absolute velocity with which they move, and
consequently the energy, but not the nature of the sensation which
they excite, which must depend, according to every analogy, upon
the duration of the oscillations. It is thus that the nature of the
sounds, transmitted by the air to our ears, depends entirely on the
duration of each of the oscillations executed by the air, or by the
sonorous [p121] body which puts it in motion; and that the greater
or less amplitude or energy of the oscillations only augments or
diminishes the intensity of the sound, without changing its nature,
that is, its tone, or pitch.
The intensity of the light must depend then on the intensity of
the vibrations of the ether; and its nature, that is to say, the
sensation of colour that it produces, will depend on the duration
of each oscillation, or on the length of the undulation, the one of
these being proportional to the other. [We find, however, nothing in
light of the same colour that is at all analogous to the different
register, quality, or _timbre_ of a sound; by which, for instance,
the sound of a violin differs from that of a flute in unison with it:
the subordinate, or harmonic tones of the sound having nothing in
light to correspond with them. TR.]
The duration of the elementary oscillation remaining the same, the
absolute velocity of the ethereal particles, at the corresponding
periods of the oscillatory motions, is, as we have seen, proportional
to its extent. It is the square of this velocity, multiplied by the
density of the fluid, that represents what is called the living force
in mechanics, or otherwise the energy or impetus of the particles,
which is to be taken as the measure of the sensation produced, or of
the intensity of the light: thus, for example, if in the same medium,
the amplitude of the oscillation is doubled, the absolute velocities
will also be doubled, and the living force, or the intensity of the
light, will be quadrupled.
We must, however, take care not to confound this absolute velocity
of the particles of the fluid with the velocity of the propagation
of the agitation. The first varies according to the amplitude of
the oscillations; the second, which is nothing but the promptitude
with which the motion is communicated from one stratum to the other,
is independent of the intensity of the vibrations. It is for this
reason, that a weak sound is transmitted by the air with the same
velocity as a stronger one; and that the least intense light is
propagated with the same rapidity as the brightest. When we speak
of the velocity of light, we always speak of the velocity of its
propagation. Thus, when we say that light passes through 200 thousand
[p122] miles in a second, we do not mean, according to the undulatory
system, that such is the absolute velocity of the ethereal particles;
but that the motion communicated to the ether employs only a second
to pass to a stratum at the distance of 200 thousand miles from its
origin.
In proportion as the undulation becomes more distant from the centre
of agitation, the motion, spreading over a greater distance, must be
weakened in every part of the wave. It is shown by calculation, that
the amplitude of the oscillatory motion, or the absolute velocity
of the particles concerned in it, is inversely proportional to the
distance from the centre of agitation. Consequently, the square
of this velocity is inversely proportional to the square of the
distance, and the intensity of the light must be inversely as the
square of the distance from the luminous point. It must be remarked,
that, for the same reasons, the sum of the living forces of the
whole undulation remains unaltered; for, on one side the length
of the undulation _d_, which may also be called its thickness, is
invariable, and its extent of surface augmenting in proportion to
the square of the distance from the centre, the quantity, or mass of
the fluid agitated, is proportional to the same square: and since
the squares of the absolute velocities are diminished in the same
proportion as the masses have augmented, it follows that the sum of
the products of the masses by the squares of the velocities, that
is to say, the sum of the living forces, remains unaltered. It is a
general principle of the motion of elastic fluids, that however the
motion may be extended or subdivided, the total sum of the living
forces remains constant; and this is the principal reason why the
living force must be considered as the measure of light, of which the
total quantity always remains very nearly the same, at least as long
as it continues to pass through perfectly transparent mediums.
It may be remarked, that black substances, and even the most
brilliant metallic surfaces, by no means reflect the whole of the
light which falls on them; bodies which are imperfectly transparent,
and even the most transparent, when of great thickness, absorb also,
to use a common expression, a considerable portion of the light that
is passing through [p123] them: but it must not be inferred that
the principle of living forces is inapplicable to these phenomena;
it follows, on the contrary, from the most probable idea that can
be formed of the mechanical constitution of bodies, that the sum
of the living force must remain always the same, as long as the
accelerating forces tending to bring the particles to their natural
positions remain unchanged, and that the quantity of living force
which disappears in the state of light, instead of being annihilated,
is reproduced in the form of heat.
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. Such a
series of regular and uninterrupted luminous motions I call a _system
of undulations_.
It is natural to suppose, on account of the prodigious rapidity
of the vibrations of light, that the luminous particles may
perform a great number of regular oscillations in each of the
different mechanical situations in which they are placed during
the combustion or the incandescence of the luminous body, although
these circumstances may still succeed each other in extremely short
periods; for the millionth part of a second is sufficient to exhibit,
for example, 545 millions of undulations of yellow light; so that
the mechanical disturbances, which derange the regular succession
of the vibrations of the luminous particles, or which even change
their nature, might be repeated a million times in a second without
preventing the regular succession of more than 500 millions of
consecutive undulations in each state of the particle. We shall soon
have occasion to apply this observation to the determination of the
circumstances in which the interference of luminous waves is capable
of producing sensible effects.
We have seen that each undulation produced by an oscillatory motion
was composed of two semiundulations, which [p124] occasioned in the
particles of the fluids velocities exactly equal in their intensity,
though opposite in the direction of the motions. Let us at first
suppose that two whole undulations, moving in the same line and in
the same direction, differ half an undulation in their progress:
they will then be superinduced on each other through one half of
their length, or of their breadth, as we should say in speaking
of the waves of a liquid: but I here use in preference the term
length as applied to the interval between the two points which are
similarly affected by the motions of two consecutive undulations.
In the supposed case of the coincidence of one half of each of the
undulations, the interference will only take place with respect to
the parts so coinciding: that is, to the latter half of the first
undulation, and the preceding half of the second: and if these two
semiundulations are of equal intensity, since they tend to give, to
the same points of the ether, impulses directly opposite, they will
wholly neutralise each other, and the motion will be destroyed in
this part of the fluid, while it will subsist without alteration in
the two other halves of the undulations. In such a case, therefore,
half of the motion only would be destroyed.
If now we suppose that each of these undulations, differing in their
progress by half the whole length of each, is preceded and followed
by a great number of other similar undulations; then, instead of
the interference of two detached undulations, we must consider the
interference of two systems of waves, which may be supposed equal
in their number and their intensity. 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 these
semiundulations will always constitute a very small part of the whole
series to be considered.
This reasoning is obviously applicable to such systems only [p125]
as are composed of undulations of the same length; for if the waves
were longer one than the other, however small their difference might
be, it would happen at last that their relative position would not
be the same throughout the extent of the groups; and while the
first destroyed each other almost completely, the following ones
would be less in opposition, and would ultimately agree completely
with each other: hence there would arise a succession of weak and
strong vibrations analogous to the beatings which are produced by
the coincidence of two sounds differing but little from each other
in their tone; but these alternations of weaker and stronger light,
succeeding each other with prodigious rapidity, would produce in the
eye a continuous sensation only.
It is very probable that the impulse of a single luminous
semiundulation, or even of an entire undulation, would be too weak
to agitate the particles of the optic nerve, as we find that a
single undulation of sound is incapable of causing motion in a body
susceptible of a sympathetic vibration. It is the succession of the
impulse, which, by the accumulation of the single effects, at last
causes the sonorous body to oscillate in a sensible manner; in the
same manner as the regular succession of the single efforts of a
ringer is at last capable of raising the heaviest church bell into
full swing. Applying this mechanical idea to vision, supported as it
is by so many analogies, we may easily conceive that it is impossible
for the two remaining semiundulations, which have been mentioned, to
produce any sensible effect on the retina; and that the result of
such a combination of the two systems must be the production of total
darkness.
If again we suppose the second system of undulations to be again
retarded half an undulation more, so as to make the difference of the
progress an entire undulation, the coincidence in the motions of the
two groups will be again restored, and the velocities of oscillation
will conspire and be augmented in the points of superposition; the
intensity of the light being then at its maximum.
Adding another semiundulation to the difference in the progress of
the two systems, so as to make it an interval and [p126] a half, it
is obvious that the semiundulations, superinduced on each other,
will now possess opposite qualities, as in the case of the half
interval first supposed: and that all the undulations must in this
manner be neutralised, except the extreme three semiundulations on
each side, which will be free from interference. Thus almost the
whole of the motion will again be destroyed, and the combination of
the two pencils of light must produce darkness, as in the case first
considered.
Continuing to increase the supposed difference by the length of a
semiundulation at each step, we shall have alternately complete
darkness and a maximum of light, accordingly as the difference
amounts to an odd or an even number of semiundulations: that is,
supposing always that the systems of undulations are of equal
intensity: for if the one series were less vivid than the other, they
would be incapable of destroying them altogether: the velocities of
the one series would be subtracted from those of the other, since
they would tend to move the particles of the ether in contrary
directions, but the remainders would still constitute light, though
feebler than that of the strongest single pencil. Thus the second
pencil would still occasion a diminution of the light: but the
diminution would be the less sensible as the pencil is supposed to be
weaker.
Such are the consequences of the principle of the interference
of undulations, which agree perfectly, as we have seen, with the
law of the mutual influence of the luminous rays which is deduced
from experiment: for the results are expressed precisely in the
same words, if we give the name of _length of undulation_ to the
difference of routes which had been represented by the symbol _d_.
Admitting, therefore, as there is every reason to believe, that light
consists in the undulations of a subtile fluid, the period _d_, after
which the same effects of interference are repeated, must be the
length of an undulation.