Theoretical Framework for Initial Transient and Steady-State Frequency Amplitudes of Musical Instruments as Coupled Subsystems
Rolf Bader · 2010
A theoretical framework for initial transients of musical instruments is presented for string-body, reed-air column and bow-string interactions. The time dependent amplitudes of the frequencies present in the two coupled systems are written as sums of phase relations of complex amplitudes of both, the system itself and the coupled system. For the steady-state parts some terms cancel because of their limits summing to zero explaining the take-over of one system by the other, like the forced oscillation of a violin body by its string frequencies. For the initial transients the equation systems consist of additional terms explaining the complex time-dependent behaviour of the initial transients. These terms formulate the temporal development of the amplitudes of additional frequencies during the transients. This enables a simple and real-time simulation of the transient time series. Additionally, two main reasons are found for the slaving of one system by the other in the discussion of zero limit sums of the steady-state phase, which are the difference in damping of the two systems and the difference in dimensionality. So e.g. the one-dimensional string with only two reflection points and low damping is much more capable to force the body to the strings frequencies as the three-dimensional body with strong damping and a complex geometry with multiple wave reflection points.