ON THE PROBLEM OF RANDOM VIBRATIONS, AND OF RANDOM FLIGHTS IN ONE, TWO, OR THREE DIMENSIONS.
[Philosophical Magazine, Vol. xxxvil. pp. 321—347, 1919.]
WHEN a number (n) of isoperiodic vibrations of unit amplitude are combined, the resultant depends upon the values assigned to the individual phases. When the phases are at random, the resultant amplitude is indeterminate, and all that can be said relates to the probability of various amplitudes (r\ or more strictly to the probability that the amplitude lies within the limits r and r + dr. The important case where n is very great I considered a long time ago* with the conclusion that the probability in question is simply
-e-^rdr .................................. (1)
The phase (0) of the resultant is of course indeterminate, and all values are equally probable.
The method then followed began with the supposition that the phases of the unit components were limited to 0° and 180°, taken at random, so that the points (r, 9), representative of the vibrations, lie on the axis 0 = 0, and indifferently on both sides of the origin. The resultant as, being the difference between the number of positive and negative components, is found from Bernoulli's theorem to have the probability
The next step was to admit also phases of 90° and 270°, the choice between these two being again at random. If we suppose %n components at random along ±30, and.|?i also at random along ±y, the chance of the representative point of the resultant lying within the area dcsdy is evidently
* Phil. Mag. Vol. x. p. 73 (1880); Scientific Payer?, Vol. i. p. 491. t See below.m's experiments.