Wave propagation on helices and hyperhelices: a fractal regression

A hyperhelix of order N is defined to be a self-similar object consisting o f a thin elastic rod wound into a helix, which is itself wound into a large r helix, until this process has been repeated N times. Wave propagation on such a structure can be discussed in a hierarchical manner, ultimately in t erms of the wavenumber k defining propagation on the elementary rod. It is found that the dispersion curve expressing the wave frequency omega as a fu nction of the elementary wavenumber k on the rod making up the initial heli x is also a fractal object, with all the macroscopically observable wave ph enomena for a hyperhelix of arbitrarily large order being compressed into a small wavenumber range of width about 2R(2)(-1)alpha centred on the value k = R-1(-1), where R-1 is the radius, ct is the helical pitch angle of the smallest helix in the progression, and R-2 is the radius of the next-larger helix.