Vibration of a Bowed String : Application of Timoshenko's Beam Theory and Transmission of Sound Energy Through the Bridge

The bow force necessary to maintain steady vibration of a bowed string is greatly increased, and its fluctuations are severely intensified by the presence of inharmonicity due to flexural stiffness of the string. Timoshenko's beam theory gives a slightly smaller estimation of inharmonicity than the ordinary beam theory and decreases the necessary bow force considerably. Meanwhile, the internal friction of the string in bending and torsional motions and the compliance and damping of the bow mitigate the effect of inharmonicity that comes from the stiffness of the string. These factors are related with the configurations of the velocity resonance curve of the string, especially of its bottom parts in the higher frequency range. Along with some of our previous reports, the sound energy transmitted through the bridge to the body of an instrument is considered in this analysis, in which a Fourier series solution by means of series transformation for a piecewise-linear system is used.