16-3 Energy and Power of a Wave Traveling Along a String#

Prompts

As a wave travels along a string, where is the energy stored? Is it kinetic, potential, or both?
At what position (\(y = 0\) or \(y = \pm y_m\)) does a string element have maximum kinetic energy? Maximum elastic potential energy? Why?
How does the wave transport energy along the string? What role does the tension play?
Write the formula for the average power \(P_{\text{avg}}\) transmitted by a sinusoidal wave. What does it depend on (\(\mu\), \(v\), \(\omega\), \(y_m\))?
If you double the amplitude \(y_m\) of a wave, by what factor does the average power change? If you double the frequency \(f\)?

Lecture Notes#

Overview#

A wave on a string transports energy as it travels. The energy is stored in the oscillating string elements as kinetic and elastic potential energy.
Kinetic energy peaks when an element rushes through equilibrium (\(y = 0\)); potential energy peaks there too (maximum stretch). Both are zero at the extremes (\(y = \pm y_m\)).
The average power \(P_{\text{avg}}\) is the rate at which energy is transmitted; \(P_{\text{avg}} \propto \omega^2 y_m^2\)—doubling amplitude quadruples power.

Kinetic and potential energy#

As a sinusoidal wave passes, each string element undergoes SHM (see section 16-1).

Kinetic energy: \(dK = \frac{1}{2}(dm)\,u^2\), where \(u\) is the transverse velocity.

Maximum at \(y = 0\)—element rushing through equilibrium.
Zero at \(y = \pm y_m\)—element momentarily at rest at extremes.

Elastic potential energy: The wave stretches the string; PE is associated with length changes (like a spring).

Maximum at \(y = 0\)—element has maximum stretch when crossing equilibrium.
Zero at \(y = \pm y_m\)—element has its undisturbed length at the extremes.

Energy location

Unlike a free oscillator, a string element in a wave does not have constant total energy. At \(y = 0\): both KE and PE are maximum—the element receives energy from the wave. At \(y = \pm y_m\): both are zero (element at rest, string unstretched)—the element has passed that energy along. Energy flows through each element as the wave propagates.

Energy transport#

The wave transports energy along the string. Tension forces do work, transferring energy from regions that have it (near \(y = 0\)) to regions that do not (near \(y = \pm y_m\)). As the wave advances, energy flows in the direction of propagation.

Average power#

The average power—average rate at which energy is transmitted—for a sinusoidal wave on a string is

(97)#\[ P_{\text{avg}} = \frac{1}{2} \mu v \omega^2 y_m^2 \]

\(\mu\) (linear density) and \(v\) (wave speed): depend on the string (material, tension).
\(\omega\) (angular frequency) and \(y_m\) (amplitude): depend on the source.

Important

\(P_{\text{avg}} \propto \omega^2\) and \(P_{\text{avg}} \propto y_m^2\). Doubling the amplitude quadruples the power; doubling the frequency also quadruples the power. This \(\omega^2\) and \(y_m^2\) dependence is general for waves of all types.

Summary#

KE maximum at \(y = 0\); PE maximum at \(y = 0\) (both zero at \(y = \pm y_m\)).
The wave transports energy; tension transfers it along the string.
\(P_{\text{avg}} = \frac{1}{2}\mu v \omega^2 y_m^2\)—power scales as \(\omega^2\) and \(y_m^2\).