Chapter 15, Problem 85E

(a)

To determine

To show: The kinetic energy per unit length of a string for a wave on the string is $u_k\left(x,t\right)=\frac{1}{2}\mu v_y^2\left(x,t\right)=\frac{1}{2}\mu {\left(\frac{\partial y\left(x,t\right)}{\partial t}\right)}^2$ .

(b)

To determine

The value of $u_k\left(x,t\right)$ for the sinusoidal wave on the string.

(c)

To determine

To show: The stretched length of the line segment is approximately $\Delta x\left[1+\frac{1}{2}{\left(\frac{\partial y\left(x,t\right)}{\partial x}\right)}^2\right]$.

(d)

To determine

The work done and show that the potential energy per unit length of the string is $u_p\left(x,t\right)=\frac{1}{2}F{\left(\frac{\partial y}{\partial x}\right)}^2$.

(e)

To determine

The value of $u_p\left(x,t\right)$ for the sinusoidal wave on the string.

(f)

To determine

To show: The kinetic and potential energies are equal for all $x$ and $t$ for the sinusoidal wave.

(g)

To determine

To show: The displacement $y\left(x,t\right)$ and the energies $u_k\left(x,t\right)$ and $u_p\left(x,t\right)$ in a graph and explain why $u_k\left(x,t\right)$ and $u_p\left(x,t\right)$ are maximum where $y\left(x,t\right)$ is zero and vice versa.

(h)

To determine

To show: The instantaneous power in the wave is equal to the total energy per unit length multiplied by the wave speed $v$.