Nonlinear generation of missing modes on a vibrating string

Abstract

The nonlinear transfer of energy among modes of different frequencies on a vibrating string is investigated both theoretically and experimentally. The nonlinearity is associated with the well-known variation of string tension caused by the vibration modes, but it is essential that at least one of the end supports has finite mechanical admittance if there is to be any mode coupling. If the nonrigid bridge support has zero admittance in a direction parallel to the string, the coupling is of third order in the mode amplitudes. For a more realistic model in which the string changes direction as it passes over a bridge of finite admittance there are additional coupling terms of second order. The first mechanism gives driving terms of frequency 2ωn±ωm where ωn and ωm are, respectively, the angular frequencies of the nth and mth modes present on the string, while the second mechanism gives driving terms of angular frequencies 2ωn and 2ωm. Analysis shows that modes absent from the initial excitation of the string can be driven to appreciable amplitude by these mechanisms, reaching their maximum amplitude after a time typically of order 0.1 s. Modes that are in nearly harmonic frequency relationship behave simply but coupling of modes that are appreciably inharmonic may give rise to rapid amplitude fluctuations. A simple experiment with a wire deflected by a bridge of elastic cord and plucked so as to eliminate a particular mode from the initial excitation provided general semiquantitative confirmation of the theoretical predictions.