Wave propagation on helices and hyperhelices: a fractal regression

A hyperhelix of order N is de­ ned 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 k de­ ning propagation on the elementary rod. It is found that the dispersion curve expressing the wave frequency ! as a function of the elementary wavenumber k 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 2R1 2 ¬ centred on the value k = R ¡1 1 , where R1 is the radius, ¬ is the helical pitch angle of the smallest helix in the progression, and R2 is the radius of the next-larger helix.