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String Slope from Velocity Waves

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

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

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 $ +1/c$. 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). $

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Use of the Chain Rule