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From 1983 to 1995, Richard H. Lyon published several papers on Statistical Phase Analysis, showing that the average phase of the transfer functions in complex systems grows with frequency in proportion to the modal density of the system. In one dimensional systems this phase growth is the same as that of freely propagating waves. However, in two and three dimensional systems this phase growth is much larger than the corresponding freely propagating wave. Recent work has shown that these phase growth functions can be used as mode shape functions in discrete system models to obtain results consistent with Statistical Energy Analysis. This paper reviews these results and proposes naming the statistical mode shape functions in honor of Lyon.

A new formulation for dynamic analysis of the response of vibro-acoustic systems is developed. The method is based on a discrete element formulation similar in geometry to a finite element model. However, the Dynamic Element Analysis uses transcendental functions for the response interpolation functions. The phase of the functions converges at high frequencies to the Statistical Phase. At low frequencies the interpolation functions converge to the polynomials used in finite elements. Thus, the Dynamic Element Analysis covers a wide frequency range without requiring a refinement of the mesh, and it provides a deterministic response in the mid-frequency range before converging to a statistically correct response at high frequencies. Examples are shown of the response of structures and acoustic radiation.