On the Partials of a Pianoforte String Struck by an Elastic Hammer
R. N. Ghosh · Physical Review · 1924
Dynamics of a stretched string struck by an elastic hammer.---Neglecting small terms in the partial differential equation of motion, a solution is obtained involving the mass, elasticity and impact speed of the hammer and the distance of impact $\ensuremath{\alpha}$ from the nearer fixed end. Also approximate expressions are obtained for the pressure exerted on the string and for the duration of impact. The final solution for the displacement contains besides the Helmholtz term, three others. For a given $\ensuremath{\alpha}$, the Helmholtz term disappears for partials with periods 2, 2/3, 2/5, etc., but not the other terms; this explains the observed existence of these partials with feeble intensity. The equation shows how the relative strengths of the partials depend on the various factors. If the duration of impact is half the period of vibration of a component, that component will become very prominent. For a given velocity of impact, the resulting amplitude increases with $\ensuremath{\alpha}$.