Wave Propagation on Helices and Hyperhelices: A Fractal Regression

Abstract

A hyperhelix of order N is defined to be a self-similar object consisting of a thin elastic rod wound into a helix, which is itself wound into a larger helix, until this process has been repeated N times. Wave propagation on such a structure can be discussed in a hierarchical manner, ultimately in terms of the wavenumber κ defining propagation on the elementary rod. It is found that the dispersion curve expressing the wave frequency ω as a function of the elementary wavenumber κ on the rod making up the initial helix is also a fractal object, with all the macroscopically observable wave phenomena for a hyperhelix of arbitrarily large order being compressed into a small wavenumber range of width about 2R2-1α centred on the value κ = R1-1, where R1 is the radius, α is the helical pitch angle of the smallest helix in the progression, and R2 is the radius of the next-larger helix.