Mersenne's law were first proposed by Marin Mersenne in his 1636 work, Harmonie Universalis. It described the frequency of oscillation of a stretched string in relation to its length, its density and the stretching force and gave an equation for the fundamental frequency. This was all established on the basis of experiments and not on the basis of mechanics which was only then beginning to gather momentum with the work of Galileo, Huygens and Newton with Newton publishing his Principia in 1687.
Although Mersenne's law was correct, his measurements were not very precise and they were greatly improved by Joseph Saveur. Incidentally, he is credited with coining the term acoustique which he derived from the Greek word meaning able to be heard. Moreover, in 1701 he proposed the superposition principle. Again, all of this work was experimental with no theoretical mechanics involved.
Brook Taylor of Taylor series fame was a fervant Newtonian and probably one of mathematics most artistically gifted members, he learned in his adolescence to play the harpsichord and to paint and which later found mathematical expression through his pioneering theoryof the vibrating string and also of perspective. He was the first mathematician to deduce Mersenne's expression from the geometry of the string and its mechanics by considering it as a row of rigid particles.
His interest in oscillations is apparent in his first important paper which was published in The Philosophical Transactions of the Royal Society in 1714 but which was actually written in 1708, according to his correspondance with John Keill, a mathematician at St John's, Oxford, where he had matriculated in 1701. Here he tackled the centre of oscillation of a body. The work led to a priority dispute with Bernouilli.
The following year he published Methodus Incrementorum Directa et Inversa in London. Here, he moves on to the theory of the transverse vibrations of strings. He wasn't attempting to find the equation of motion for a string, but was considering the oscillations of a flexible string in terms of the isochrony of a pendulum. He showed that the resultant of forces acting on the extremities of a string was proportional to the curvature and directed along its normal .
Newton had approached curvature as the determination of the centre of curvature as the limit of the intersection of two normals. Although this was not published until 1736, Taylor was familiar with this work but preferred, at least here, to concieve the radius of curvature as Newtons continental rival, Liebniz, did, taking it to be centre of a limiting circle through three points and associated curvature with the angle of contact dating back to Euclid.
He then used these notions to describing the fundamental mode of a vibrating string. In propositions XXII & XXIII he showed that under his conditions each point would vibrate as a cycloidal pendulum, and determined the period in terms of the length of the string and its tension. In deriving this formula he assumed that every point of the string passes through its equilibrium position at the same time, an assumption that D'Alembert showed later to be unnecessary.
Although Taylor was also able to describe the form of the string at any instant it was nevertheless a static solution, the element of time was not really introduced. This was the achievement of D'Alembert who gave the equation of motion of the string as the one dimensional wave equation in 1747 in his Recherches sur la courbe qui forme une corde tendue mise en vibration in which he obtained the PDE of a vibrating string by combining the restoring force of Taylor's with Newton's acceleration law.
It's probably worth adding that in 1751, David Bernouilli gave a theoretical justification of the superposition principle that had been proposed by Joseph Saveur in 1701 and that in the late 1740s, a bitter dispute broke out between d'Alembert and Euler over the solution of the wave equation. The issues were, the nature of mathematical functions, that is, what is meant by their continuity and differentiability and which highlighted the relations between mathematics, metaphysics and experimental natural philosophy in the 18C.
Elizabeth Garber, The Language of Physics