The Random Vibrations of a String*

where the qk represent charges, Lik inductances, Rik resistances, Gik reciprocals of capacitance, the Ejk are random e.m.f's**, and primes denote differentiations with respect to time. The theory of such systems of equations, quite carefully examined during the war years in applications to noise in electrical networks, turns out also to be applicable to the various mechanical systems—in particular, it is applicable to the system of a vibrating string with fixed end points. A general theory for the system of equations (1) has been developed by Uhlenbeck and Wang [2], Some of the results which we shall obtain in this article have been derived without making use of the general theory. We shall make comparison with these results and in addition we shall derive several more results. The results of Uhlenbeck and certain of his co-authors [1, 5], have been derived directly from the differential equation of motion for the string. In order to apply the general theory to the vibrating string, it is necessary to discretize the string. We therefore assume that the stringf is made up of n + 2 particles, (2 fixed, n vibrating) of equal mass m harmonically bound together by means of massless springs. Furthermore let us assume that this system of particles has random forces acting on it and that as a result the system vibrates, the vibration taking place in a plane. As a last assumption we suppose that the vibration takes place in a viscous medium so that each of the particles undergoes a damped vibration. When we have obtained our results for the discretized system, we can derive the results for the case of a continuous string by a limiting procedure, namely that of letting n the number of vibrating particles go to infinity while the total mass and length of the system remains constant. In this article we shall derive the following: 1) the characteristic function of the sum of the squares of the deviations of the displacement of the particles from their given initial positions—also the corresponding characteristic function for the continuous string,