(a) Show that for a wave on a string, the kinetic energy per unit length of string is $$u_{\mathrm{k}}(x, t)=\frac{1}{2} \mu v_{y}^{2}(x, t)=\frac{1}{2} \mu\left(\frac{\partial y(x, t)}{\partial t}\right)^{2}$$

where $\mu$ is the mass per unit length. (b) Calculate $u_{\mathrm{k}}(x, t)$ for a sinusoidal wave given by Eq. (15.7). (c) There is also elastic potential energy in the string, associated with the work required to deform and stretch the string. Consider a short segment of string at position $x$ that has unstretched length $\Delta x,$ as in Fig. $15.13 .$ Ignoring the (small) curvature of the segment, its slope is $\partial y(x, t) / \partial x$ . Assume that the displacement of the string from equilibrium is

small, so that $\partial y / \partial x$ has a magnitude much less than unity. Show that the stretched length of the segment is approximately $$\Delta x\left[1+\frac{1}{2}\left(\frac{\partial y(x, t)}{\partial x}\right)^{2}\right]$$ (Hint: Use the relationship $\sqrt{1+u} \approx 1+\frac{1}{2} u,$ valid for $|u|<<$

1.) (d) The potential energy stored in the segment equals the work done by the string tension $F$ (which acts along the string) to stretch the segment from its unstretched length $\Delta x$ to the length calculated in part (c). Calculate this work and show that the potential energy per unit length of string is $$u_{\mathrm{p}}(x, t)=\frac{1}{2} F\left(\frac{\partial y(x, t)}{\partial x}\right)^{2}$$

(e) Calculate $u_{\mathrm{p}}(x, t)$ for a sinusoidal wave given by Eq. $(15.7) .$ (f) Show that $u_{\mathrm{k}}(x, t),=u_{\mathrm{p}}(x, t),$ for all $x$ and $t .(\mathrm{g})$ Show $y(x, t)$ $u_{\mathrm{k}}(x, t),$ and $u_{\mathrm{p}}(x, t)$ as functions of $x$ for $t=0$ in one graph with

all three functions on the same axes. Explain why $u_{\mathrm{k}}$ and $u_{\mathrm{p}}$ are

maximum where $y$ is zero, and vice versa. (h) Show that the instantaneous power in the wave, given by Eq. $(15.22),$ is equal to the total energy per unit length multiplied by the wave speed $v$

Explain why this result is reasonable.