XIX. A problem on the summation of simple harmonic functions of the same, amplitude and frequency but of random phase

Abstract

Abstract The following problem has arisen in the theoretical study of site errors in direction finding. “A number of simple harmonic functions of the same amplitude and frequency, but having phase angles which may have any equally probable value between zero and 2φ, are added together. That component of the resultant simple harmonic function is found which has an arbitrarily chosen phase. What is the probability that the amplitude of this component will lie between any assigned limits?” The problem is similar to that of the. “Random Walk,” to which solutions have been given by previous workers, and a brief resume is given of the Random Walk problem and its solution. The solution of the present problem is discussed in terms of the probability function Pn(s), where Pn(s)ds is the probability that the amplitude of the chosen component will lie between s and s+ds for a given number n of harmonic functions. It is shown that Pn(s) is not in general expressible in terms of well-known functions, and it must then ...