In 1746 the work was extended by Jean Le Rond d’Alembert in
He derived a difference equation for the displacement of the kth mass as a function of time. In 1746 the work was extended by Jean Le Rond d’Alembert in his attack on the vibrating string problem typified by the violin string. To treat the continuous string d’Alembert increased the number of masses, letting their number go to infinity, while the size and mass of each individual “bead” went to zero, such that the mass density of the string (mass per unit length) remained finite and constant over the string. In this manner d’Alembert derived the equation.
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