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Early Musical Acoustics

All things which can be known have number; for it is not possible that without number anything can be either conceived or known.
-- Philolaus (ca. 400 BC)

Vibrating strings were studied by the Pythagoreans (6th-5th century BC). Pythagorus noticed that harmonics were produced by dividing the string length by whole numbers, and he realized (ca. 550 BC) that the vibrations produced in air were at the same frequency as that of the string [327]. ``Harmony theory'' from the Pythagoreans was taught throughout the Middle Ages as one of the seven liberal arts: the quadrivium, consisting of arithmetic, geometry, astronomy, and music (harmony theory); and the trivium, consisting of grammar, logic, and rhetoric [385].

It took until Galileo (1564-1642) to be free of the formulation of Aristotle (384-322 BC) that all motion required an ongoing applied force, thereby opening the way for modern differential equations of motion. The ideas of Galileo were formalized and extended by Newton (1642-1727), whose famous second law of motion ``$ f=ma$'' lies at the foundation of essentially all classical mechanics and acoustics. Newton's Principia (1686) describes sound as traveling pressure pulses, and single-frequency sound waves were analyzed.

The first to publish a one-dimensional wave equation for the vibrating string was the applied mathematician Jean Le Rond d'Alembert (1717-1783) [93].D.1The 1D wave equation can be written as

$\displaystyle K\frac{\partial^2 y}{\partial x^2} = \epsilon\frac{\partial^2 y}{\partial t^2}$ (D.1)

where

\begin{displaymath}\begin{array}{rclrcl} K & \isdef & \mbox{string tension} & \q...
...r mass density} & & = & \mbox{string displacement}, \end{array}\end{displaymath}    

$ t$ is time in seconds, and $ x$ denotes position along the string. (See discussion in §G.1.) This is thought to be the first publication of any partial differential equation as well. It can be derived directly from Newton's second law applied to a differential string element. In addition to introducing the 1D wave equation, d'Alembert introduced its solution in terms of traveling waves:

$\displaystyle y(t,x) = y_r\left(t-\frac{x}{c}\right) + y_l\left(t+\frac{x}{c}\right)$ (D.2)

where $ c=\sqrt{K/\epsilon}$ denotes the wave propagation speed. Earlier, in 1717, Brook Taylor (1685-1731)D.2published what was apparently the first mathematical paper on vibrating strings, but without introducing differential equations. D'Alembert's ideas were developed into essentially modern form primarily by Euler (1707-1783)D.3 [62]. Incidentally, d'Alembert also invented the theory of construction of eyeglasses.D.4


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[How to cite and copy this work] 
``Physical Audio Signal Processing for Virtual Musical Instruments and Digital Audio Effects'', by Julius O. Smith III, (December 2005 Edition).
Copyright © 2006-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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