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The use of ensemble averaging procedures in studying the dynamics of structurally irregular mechanical systems is discussed. It is found that taking a linear average over the ensemble does not always yield the behaviour of a “typical” member of that ensemble. Under certain circumstances there are occasional members with properties so exceptional as to weight the average appreciably, and the linear ensemble average can then be seriously misleading. The effect is illustrated by the response and transmission properties of two simple mechanical models, a system of coupled oscillators with varying natural frequencies, and a vibrating string or bending beam with irregularly spaced mass and spring constraints. The distinction between the ensemble average and the typical value is important in understanding the phenomenon of Anderson localization described previously [1–3] by using these two models. This paper provides some analytic results, while an accompanying paper [4] provides complementary results from a numerical study of the coupled-oscillator model.

Confinement of vibration by one-dimensional disorder, I: Theory of ensemble averaging