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

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

By linearity of differentiation, the string slope is given by

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

Consider only the right-going component, and define

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

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

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

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

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