String Slope from Velocity Waves

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Let's use the above result to derive the slope of the ideal vibrating string From Eq. $\,$ (H.11), we have the string displacement given by

$\displaystyle y(t,x) = y_r(t-x/c) + y_l(t+x/c).$

By linearity of differentiation, the string slope is given by

$\displaystyle s(t,x) \isdef \frac{\partial}{\partial x} y(t,x) = \frac{\partial}{\partial x}y_r(t-x/c) + \frac{\partial}{\partial x}y_l(t+x/c).$

Consider only the right-going component, and define

$\displaystyle {\dot y}_r(\tau) \isdef \frac{d}{d\tau} y_r(\tau)$

with $\tau\isdef t-x/c$ . By the chain rule,

$\displaystyle \frac{\partial}{\partial x}y_r(t-x/c) = \frac{dy_r}{d\tau} \cdot \frac{d\tau}{dx} = {\dot y}_r(t-x/c)\cdot\left(-\frac{1}{c}\right).$

The left-going component is similar, but with

. Thus, the string slope in terms of traveling velocity-wave components can be written as

$\displaystyle s(t,x) = \frac{1}{c}{\dot y}_l(t+x/c) - \frac{1}{c}{\dot y}_r(t-x/c).$

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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