The wave equation and the vibrating string controversy

1747 Number and measure method

A partial differential equation for the vibration of a stretched string and a mathematical dispute about what is to count as a function at all.

The Frenchman Jean d'Alembert wrote the equation in 1747 and solved it in general form through two arbitrary functions. The Swiss Leonhard Euler allowed as a solution a curve drawn free-hand, and Daniel Bernoulli gave the solution as a sum of sines; d'Alembert insisted on a single analytic expression for the whole string.

The dispute over admissible functions took several decades and prepared both the generalisation of the concept of function itself and the expansion of arbitrary functions in trigonometric series.

Why this date

The date taken is 1747: d'Alembert's memoir belongs to the Berlin Academy volume "for the year 1747" (vol. 3, pp. 214-219), yet the volume itself was printed later, and catalogues differ between 1749 and 1750. The dispute went on afterwards: Euler's reply in 1748, Daniel Bernoulli's series solution in 1753. In print this is the first appearance of the wave equation specifically, and not the first partial differential equation at all: Cajori dates clear examples of such equations to 1719 in Nicolaus Bernoulli and to Euler's work of 1734.

What it grew from

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