FIRST APPEARANCE OF THE WAVE EQUATION.


D'ALEMBERT, LEONHARD EULER, DANIEL BERNOULLI. - THE CONTROVERSY ABOUT VIBRATING STRINGS.

Collection of the 9 groundbreaking papers clarifying the problems of Vibrating Strings by the use of partial differential equations. (1-3 by D'Alembert). (4-6 by Euler). (7-9 by Bernoulli). 1. Recherches sur la Courbe que forme une Corde tenduë mise en vibration. - 2. Suite des Recherches sur la Courbe que forme unde Corde tendüe, mise en vibration. - 3. Addition au memoire sur la Courbe que forme une Corde tendüe, mise en vibration. - 4. Sur la Vibration des Cordes. - 5. Sur le Mouvement d'une Corde qui au Commencement n'a été ébranlée que dans une Partie. - 6. Remarques sur les Memoires précedens de M. Bernoulli. (See No 7). - 7. Sur les Mélange de plusieurs Especes de Vibrations simples isochrones, qui peuvent coexister dans un meme Système de Corps. - 8. Reflexions et Eclaircissemens sur les Nouvelles Vibrations de Cordes exposées dans les Mémoires de l'Academie de 1747. & 1748. (identical with nos 1 and 4) - 9. Mémoire, sur les Vibrations des Cordes d'une Épaisseur inégale.

(Berlin, Haude et Spener, 1749-67). 4to. Bound together in one very nice recent marbled paper binding with gilt leather title-label to front board. All 9 without wrappers as extracted from "Memoires de L'Academie Royale des Sciences et Belles-Lettres", Tome III pp. 214-219 a. 1 engraved plate, Tome III pp. 220-249 a. 2 engraved plates, Tome VI pp. 355-360, Tome IV pp. 67-85 a. 1 engraved plate., Tome XXI pp. 307-335 a. 3 engraved plates, Tome IX pp. 196-222 a. 1 engraved plate, Tome IX pp. 173-195 a. 2 engraved plates, Tome IX, pp. 147-172 a. 1 engraved plate, Tome XXI pp. 281-306 a. 1 engraved plate.

All 9 papers in first editions, in the periodical form. In D'Alemberts paper the WAVE-EQUATION appeared for the first time in print, and thus he was the first person to solve the mathematical equation for a vibrating string, and in the same connection it was the first success with partial differential equations. This paper opened the series of papers which are offered here. The three authors came to different conclusions on the nature of an "arbitrary" function and its expansion in trigonometric functions, a controversy brought to a conclusion only in the 19th century by Fourier, Cauchy, Dirichlet and Riemann.
"Nowadays this is the starting point for SUPERSTRINGS, which some people call "THE THEORY OF EVERYTHING". D'Alembert could not have guessed the importance his discovery would one day have. He was interested in strings because he was interested in music. He was a friend of the composer Rameau, and once wrote a book explaining Rameau's theory of harmony. So his broad interest in both the arts and sciences led D'Alembert to an important insight. Perhaps the entire universe is made of vibrating strings obeying an equation D'Alembert was first alerted to by the sound of a harpiscord." (Andrew Crumey).

The wave equation is an important partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. It arises in such fields as acoustics, electromagnetics, and fluid dynamics.
D'Alembert thought that he had given the general solution of the problem, but Euler pointed out that there is no physical reason to require that the initial position of the string is given by a single function - D'Alembert had operated with this arbitrary function (the initial condition) and one other arbitrary function, the one that gives the shape of the travelling wave - different parts of the string could very well be described by different formulas as long as they fitted together smoothly. Moreover, the travelling wave solution could be extended to his situation. The point is that for Euler and D'Alembert every function had a graph represented a single function. Euler argued that any graph - even if not given by a function - should be admitted as a possible initial position of the string. D'Alembert did not accept Euler's physical reasoning (Euler's first paper here offered). In 1755 Daniel Bernoulli joined the argument with his paper (Bernoulli's first paper offered here). He found another form of solution for the vibrating string, using "standard waves". A standing wave is a motion of the string in which there are fixed "nodes" which are stationary; between the nodes each segment of the string moves up and down in unison. The "principal mode" is the one without nodes, the "second harmonic" is the name given to the motion with a single node at the mid-point. The "third harmonic" has two equally spaced-nodes, and so on - each "harmonic" thus corresponds to a pure tone of music.
"Just as D'Alembert had rejected Euler's reasoning, now Euler rejected Bernoulli's. First of all, as Bernoulli acknowledged, Euler himself had already found the standing wave solution in one special case. Euler's objection was to claim that the standing-wave solution was general - applicable to all motions of the string" (Davis & Hersh). D'Alembert also attacked Bernoulli's solution (see his article in the Encyclopedie).
Euler's method was more useful than D'Alemberts, but, as it turned out, Bernoulli was closer to the truth. The later use by Fourier of sines and cosines on heat flow was very similar to Bernoulli's method of studying the vibrations. "The debate about the Vibrating Strings raged throughout the 1760s and 1770s. Even Laplace entered the fray in 1779, and sided with d'Alembert. D'Alembert continued in a series of booklets, entitled Opuscules, which began to appear in 1768". (Morris Kline).


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