Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Chapters

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

Chapter Contents:

Search Physical Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

A More Complete Derivation of the String Wave Equation

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 [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 $ 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

$\displaystyle \frac{\partial \mathbf{K}}{\partial x} = \epsilon \frac{\partial^2 \mathbf{p}}{\partial t^2}$ (G.1)

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

$\displaystyle d{\bf s}$ $\displaystyle =$