On a Limiting Process Which Asymptotically Produces f ^-2 Spectral Density

Abstract

In the recent papers in which the results of the spectral analyses of roughnesses of runways or roadways are reported [1, 2, 3,4] the power spectral densities of approximately of the form f ^-2 (f: frequency) are often treated. This fact directed the present author to the investigation of the limiting process which will provide the f ^-2 form under fairly general assumptions. In this paper a very simple model is given which explains a way how the f ^-2 form is obtained asymptotically. Our fundamental model is that the stochastic process, which might be considered to represent the roughness of the runway, is obtained by alternative repetitions of roughening and smoothing. We can easily get the limiting form of the spectrum for this model. Further, by taking into account the physical meaning of roughening and smoothing we can formulate the conditions under which this general result assures that the f ^-2 form will eventually take place.

A part of the results of the present paper was announced in July 7, 1960 at the 28th annual meeting of the Japan Statistical Association.