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Wave Equation for the Vibrating String

Consider an elastic string under tension which is at rest along the $ x$ dimension. Let $ \mathbf{i}$ , $ \mathbf{j}$ , and $ \mathbf{k}$ denote the unit vectors in the $ x$ , $ y$ , and $ z$ directions, respectively. When a wave is present, a point $ \mathbf{p}=(x,0,0)$ originally at $ x$ 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 $ y$ direction. This more general treatment is adapted from [122]. An alternative clear development is given in [394].

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 $ K$ denote string tension along $ x$ 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

$\displaystyle \frac{\partial \mathbf{K}}{\partial x} = \epsilon \frac{\partial^2 \mathbf{p}}{\partial t^2}$ (B.31)

where $ d x$ 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 $ K$ 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

$\displaystyle d{\bf s}$ $\displaystyle =$ $\displaystyle \mathbf{i}(dx + d\xi) + \mathbf{j}d\eta + \mathbf{k}d\zeta$ (B.32)
  $\displaystyle =$ $\displaystyle \left[\mathbf{i}\left(1+\frac{\partial \xi}{\partial x}\right)+
\mathbf{j}\frac{\partial \eta}{\partial x} +
\mathbf{k}\frac{\partial \zeta}{\partial x}\right]dx$ (B.33)

having magnitude

$\displaystyle ds = \sqrt{\left(1+\frac{\partial \xi}{\partial x}\right)^2 + \left(\frac{\partial \eta}{\partial x}\right)^2 + \left(\frac{\partial \zeta}{\partial x}\right)^2}.$ (B.34)

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University