The theory of light / by Thomas Preston.

  • Preston, Thomas.
Date:
1901
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Licence: In copyright

Credit: The theory of light / by Thomas Preston. Source: Wellcome Collection.

Provider: This material has been provided by UCL Library Services. The original may be consulted at UCL (University College London)

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    travelling with a velocity v, for y remains unchanged if we replace t by / + T and x by its corresponding value x + vT. Otherwise thus:—If v be the velocity of propagation, the displace- ments at any two points P and P' will be the same at the times / and t' if x'-x — v{t' — t) or x'— x-vt + vt'. Hence y = (p{x, t) = (p{x', t') = (p{x - vt + vt', t'), which holds for all values of t\ and consequently for f = 0, in which case we have y = (P{vt-x). If another disturbance be running in the opposite direction its equation will he y= f{vt + x), so that if the two be superposed we will have y = <p{vt - X) +f{vt + X), which represents the general state of disturbance in a finite stretched wire when we have both direct and reflected waves. A particular solution of the equation y = (ji(vt — x) is obviously y = a sm vt-x P The value of y will be unaltered if / be increased by 27rp/v. The periodic time is therefore T = 27rp/v, but A = ^'T = Stt/). Therefore ^ • 27r_ . y = asm—{vt-x). / A The general form of is a periodic function. It has the same values at distances x, x + A, x + 2X, etc., and at times / + T, t + 2T, etc., and we know by Fourier's theorem (Thomson and Tait, Nat. Phil.) that any periodic function f{z) of period A can be expanded as a series in the form f{z) = kx sin (^^ + ^1^ +A2sin (^^ + 2) +^3 si ^^ + a3^ -f etc. Hence we may write » • 27r, , , . = . 47r, ^ , y=ki&m—{vt- Xi) + A.2 sm — {vt - x^) + etc. A A The complete vibration is therefore made up of a series of superposed simple harmonic vibrations of wave lengths A, J A, J A, etc. As we have reason to believe that waves of different periodic times ^ Hence p is the radius of a circle of circumference X.