Energy in a Standing Wave

Consider a standing wave on a string with two fixed ends, $x=0$ and $x=L$. The motion of the string can be described by the equation

$$y(x,t) = A \sin (kx) \cos (\omega t).$$

(a)

Find the kinetic energy density, $ke(x,t)$, and the potential energy density, $pe(x,t)$. Is it true that $ke(x,t)=pe(x,t)$? What is the total energy density, $e(x,t)$? Does it depend on time? Is your result consistent with conservation of energy?

(b)

Find the power $P(x,t)$.

(c)

Show that the equation

$$\frac{\partial e}{\partial t} = - \frac{\partial P}{\partial x}$$

holds for all points on the string at all times. What is the physical meaning of this equation?

(d)

The string is oscillating at the lowest fundamental frequency. In this part, you will sketch snapshots of some quantities related to the string motion at time $t_0$ such that $\omega t_0=\pi/4$. Be sure to clearly mark the axes!

• Sketch the shape of the string, $y(x,t_0)$. On your sketch, indicate with little arrows the direction in which various points on the string are moving.

• Graph the instantaneous power, $P(x,t_0)$.

(e)

In which direction is energy propagating in the left half of the string, $0<x<L/2$? What about the right half, $L/2<x<L$?

(f)

Repeat parts (d) and (e) at time $t_1$ such that $\omega t_1=3\pi/4$.