Wave Equation

The wave equation for the ideal vibrating string may be written as

where we define the following notation:

The wave equation in this form can be interpreted as a statement of Newton's second law,

on a microscopic scale. Since we are concerned with transverse vibrations on the string, the relevant restoring force (per unit length) is given by the string tension (force along the string axis) times the curvature of the string, or ; the restoring force is balanced at all times by the inertial force per unit length of the string which is equal to mass density (mass per unit length) times transverse acceleration, i.e., . See Appendix F for a review of basic physical concepts. The wave equation is derived in some detail in Appendix G.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.