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Unformatted text preview: The wave equation in one dimension Prof. Joyner 1 The theory of the vibrating string touches on musical theory and the theory of oscillating waves, so has likely been a concern of scholars since ancient times. Nevertheless, it wasn’t until the late 1700s that mathematical progress was made. Though the problem of describing mathematically a vibrating string requires no calculus, the solution does. With the advent of calculus, Jean le Rond dAlembert, Daniel Bernoulli, Leonard Euler, Joseph- Louis Lagrange were able to arrive at solutions to the one-dimensional wave equation in the eighteenth-century. Daniel Bernoulli’s solution dealt with an infinite series of sines and cosines (derived from what we now call a “Fourier series”, though it predates it), his contemporaries did not believe that he was correct. Bernoullis technique would be later used by Joseph Fourier when he solved the thermodynamic heat equation in 1807. It is Bernoulli’s idea which we discuss here as well. Euler was wrong: Bernoulli’s method was basically correct after all. Now, d’Alembert was mentioned in the lecture on the transport equation and it is worthwhile very briefly discussing what his basic idea was. The theorem of dAlembert on the solution to the wave equation is stated roughly as follows: The partial differential equation: ∂ 2 w ∂t 2 = c 2 · ∂ 2 w ∂x 2 is satisfied by any function of the form...
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- Spring '08
- Vibrating string, Partial differential equation, wave equation, Daniel Bernoulli, Joseph Fourier