Normal Vibration Frequencies of a Stiff Piano String

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<jats:p>This paper deals with the normal modes of vibration of a stiff piano string. The equations that govern the vibration of a solid string are developed along traditional lines. They are modified to apply to strings having a solid-steel core upon which are wrapped one or two copper windings. The bass strings of most pianos are made this way. Two boundary conditions are considered: namely, (1) pinned by sharp knife edges at both ends and (2) clamped at both ends. Formulas for the partial frequencies for both of these conditions are developed. The partial frequencies that are calculated by these formulas are compared to the experimental values obtained on an upright Hamilton piano. The experimental values appear to agree somewhat better with the pinned boundary condition rather than the clamped boundary condition, although the differences are not much greater than the observational error. It was found that the formula fn = nf1[(1 plus; Bn2)/(1 + B)]12 gives values of the partial frequencies that agree with the experimental ones where n is the number of the partial, f the fundamental frequency, and B a constant that can be calculated from the dimensions of the wire.</jats:p>

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