Proposition 51, Theorem 39

If pulses are propagated through a fluid, the several particles of the fluid, going and returning with the shortest reciprocal motion, are always accelerated or retarded according to the law of the oscillating pendulum.

Let AB, BC, CD, &c., represent equal distances of successive pulses, ABC the line of direction of the motion of the successive pulses propagated from A to B; E, F, G three physical points of the quiescent medium situate in the right line AC at equal distances from each other; Ee, Ff, Gg, equal spaces of extreme shortness, through which those points go and return with a reciprocal motion in each vibration; ε, ϕ, γ, any intermediate places of the same points; EF, FG physical lineolae, or linear parts of the medium lying between those points, and successively transferred into the places εϕ, ϕγ, and ef, fg. Let there be drawn the right line PS equal to the right line Ee.

Bisect the same in O, and from the centre O, with the interval OP, describe the circle SIPi. Let the whole time of one vibration; with its proportional parts, be expounded by the whole circumference of this circle and its parts, in such sort, that, when any time PH or PHSh is completed, if there be let fall to PS the perpendicular HL or hl, and there be taken Eε equal to PL or Pl, the physical point E may be found in ε.

A point, as E, moving according to this law with a reciprocal motion, in its going from E through ε to e, and returning again through ε to E, will perform its several vibrations with the same degrees of acceleration and retardation with those of an oscillating pendulum. We are now to prove that the several physical points of the medium will be agitated with such a kind of motion. Let us suppose, then, that a medium hath such a motion excited in it from any cause whatsoever, and consider what will follow from thence.

In the circumference PHSh let there be taken the equal arcs, HI, IK, or hi, ik, having the same ratio to the whole circumference as the equal right lines EF, FG have to BC, the whole interval of the pulses. Let fall the perpendiculars IM, KN, or im, kn; then because the points E, F, G are successively agitated with like motions, and perform their entire vibrations composed of their going and return, while the pulse is transferred from B to C; if PH or PHSh be the time elapsed since the beginning of the motion of the point E, then will PI or PHSi be the time elapsed since the beginning of the motion of the point F, and PK or PHSk the time elapsed since the beginning of the motion of the point G; and therefore Eε, Fϕ, Gγ, will be respectively equal to PL, PM, PN, while the points are going, and to Pl, Pm, Pn, when the points are returning. Therefore εγ or EG + Gγ - Eε will, when the points are going, be equal to EG - LN and in their return equal to EG + ln. But εγ is the breadth or expansion of the part EG of the medium in the place εγ; and therefore the expansion of that part in its going is to its mean expansion as EG - LN to EG; and in its return, as EG + ln or EG + LN to EG.

Therefore since LN is to KH as IM to the radius OP, and KH to EG as the circumference PHShP to BC; that is, if we put V for the radius of a circle whose circumference is equal to BC the interval of the pulses, as OP to V; and, ex æquo, LN to EG as IM to V; the expansion of the part EG, or of the physical point F in the place εγ, to the mean expansion of the same part in its first place EG, will be as V - IM to V in going, and as V + im to V in its return. Hence the elastic force of the point P in the place εγ to its mean elastic force in the place EG is as

…

to

…

in its going, and

… to

… in its return.

By the same reasoning the elastic forces of the physical points E and G in going are as

…

and

… to

…

and the difference of the forces to the mean elastic force of the medium as

… to

…

that is, as

…

to

…

or as HL - KN to V; if we suppose (by reason of the very short extent of the vibrations) HL and KN to be indefinitely less than the quantity V.

Therefore since the quantity V is given, the difference of the forces is as HL - KN; that is (because HL - KN is proportional to HK, and OM to OI or OP; and because HK and OP are given) as OM; that is, if Ff be bisected in Ω, as Ωϕ. And for the same reason the difference of the elastic forces of the physical points ε and γ, in the return of the physical lineola εγ, is as Ωϕ.

But that difference (that is, the excess of the elastic force of the point ε above the elastic force of the point γ) is the very force by which the intervening physical lineola εγ of the medium is accelerated in going, and retarded in returning; and therefore the accelerative force of the physical lineola εγ is as its distance from Ω, the middle place of the vibration. Therefore (by Prop. XXXVIII, Book I) the time is rightly expounded by the arc PI; and the linear part of the medium εγ is moved according to the law abovementioned, that is, according to the law of a pendulum oscillating; and the case is the same of all the linear parts of which the whole medium is compounded. Q.E.D.

Cor. Hence it appears that the number of the pulses propagated is the same with the number of the vibrations of the tremulous body, and is not multiplied in their progress.

For the physical lineola εγ as soon as it returns to its first place is at rest; neither will it move again, unless it receives a new motion either from the impulse of the tremulous body, or of the pulses propagated from that body. As soon, therefore, as the pulses cease to be propagated from the tremulous body, it will return to a state of rest, and move no more.