The bowed string: an integral equation formulation
Robert T. Schumacher, Charles J. Amick, Christopher B. Croke · The Journal of the Acoustical Society of America · 1974
The problem of the bowed string has been cast in the form of an nonlinear integral equation of the Hammerstein (Fredholm) type, and some solutions of the fixed-point problem resulting therefrom have been obtained both analytically and numerically on a digital computer. In the equation v(τ) = ∫01K(τ,τ′)F[v(τ′)]dτ′, the velocity at the bowing point, v(τ), a function of the reduced period τ, enters the integrand via the nonlinear force F(v) between bow and string. In the model we have solved, an essential assumption is that the bow presents a real, finite mechanical impedance to the string. The kernel K involves the string admittances at the bowing point. Realistic admittances are used which include the effects of string stiffness and complex terminating impedance at the bridge. The equation, and the techniques of solution we have used, should apply to a large class of multimode self-sustained oscillators, such as wind instruments. [Supported by a grant from the National Science Foundation.]