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Physical Audio Signal Processing
A More Complete Derivation of the String Wave Equation

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A More Complete Derivation of the String Wave Equation

Consider an elastic string under tension which is at rest along the dimension. Let $\mathbf{i}$ , $\mathbf{j}$ , and $\mathbf{k}$ denote the unit vectors in the , , and directions, respectively. When a wave is present, a point $\mathbf{p}=(x,0,0)$ originally at along the string is displaced to some point $\mathbf{a}=\mathbf{p}+d\mathbf{p}$ specified by the displacement vector

$\displaystyle d\mathbf{p}= \mathbf{i}\xi + \mathbf{j}\eta + \mathbf{k}\zeta.$

Note that typical derivations of the wave equation consider only the displacement $\eta$ in the

direction. This more general treatment is adapted from [121].

The displacement of a neighboring point originally at $\mathbf{q}=(x+d x,0,0)$ along the string can be specified as

$\displaystyle d\mathbf{q}= \mathbf{i}(\xi+d\xi) + \mathbf{j}(\eta+d\eta) + \mathbf{k}(\zeta+d\zeta).$

Let denote string tension along when the string is at rest, and $\mathbf{K}$ denote the vector tension at the point $\mathbf{p}$ in the present displaced scenario under analysis. The net vector force acting on the infinitesimal string element between points $\mathbf{p}$ and $\mathbf{q}$ is given by the vector sum of the force $-\mathbf{K}$ at $\mathbf{p}$ and the force $\mathbf{K}+ (\partial \mathbf{K}/\partial x)d x$ at $\mathbf{q}$ , that is, $(\partial \mathbf{K}/\partial x)dx$ . If the string has stiffness, the two forces will in general not be tangent to the string at these points. The mass of the infinitesimal string element is $\epsilon \,dx$ , where $\epsilon$ denotes the mass per unit length of the string at rest. Applying Newton's second law gives

$\displaystyle \frac{\partial \mathbf{K}}{\partial x} = \epsilon \frac{\partial^2 \mathbf{p}}{\partial t^2}$

(G.1)

where

has been canceled on both sides of the equation. Note that no approximations have been made so far.

The next step is to express the force $\mathbf{K}$ in terms of the tension of the string at rest, the elastic constant of the string, and geometrical factors. The displaced string element $\mathbf{p}\mathbf{q}$ is the vector

$\displaystyle d{\bf s}$	$\displaystyle =$	$\displaystyle \mathbf{i}(dx + d\xi) + \mathbf{j}d\eta + \mathbf{k}d\zeta$	(G.2)
	$\displaystyle =$	$\displaystyle \left[\mathbf{i}\left(1+\frac{\partial \xi}{\partial x}\right)+ \... ...artial \eta}{\partial x} + \mathbf{k}\frac{\partial \zeta}{\partial x}\right]dx$	(G.3)

having magnitude

$\displaystyle ds = \sqrt{\left(1+\frac{\partial \xi}{\partial x}\right)^2 + \le... ...\eta}{\partial x}\right)^2 + \left(\frac{\partial \zeta}{\partial x}\right)^2}.$

(G.4)

Subsections

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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.

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