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Energy Density of Waves on a String: Instructor's Guide

Main Ideas

Students' Task

Student groups are given a snapshot of a waveform and asked to determine the points where and when the kinetic energy density, potential energy density, and total energy density are maximal or minimal. Some groups consider a standing wave; other groups consider a traveling wave.

Prerequisites

Before starting this activity, through traditional lecture or the optional activities linked below, students need to acquire understandings of the solutions to the non-dispersive wave equation: $$\frac{\partial^{2}}{\partial x^{2}}\psi(x,t) = \frac{1}{v^{2}} \frac{\partial^{2}}{\partial t^{2}} \psi(x,t)$$ for traveling waves: $${\mathop{\rm Re}\nolimits} \left[ {\psi (x,t) = A{e^{i(kx \pm \omega t)}}} \right]$$ $$\psi(x,t) = A sin(kx \pm \omega t)$$ and standing waves: $$\psi(x,t) = Bsin(kx+\phi)cos(\omega t+\delta)$$

Props/Equipment

Activity: Introduction

Students need a brief introduction to the energy density of a wave on a string: $$w(x,t)=\frac{1}{2}\mu \left(\frac{\partial \psi}{\partial t}\right)^{2} + \frac{1}{2}T \left(\frac{\partial \psi}{\partial x}\right)^{2}$$ where $\mu$ is the mass density of the rope and $T$ is the tension in the rope.

Activity: Student Conversations

Activity: Wrap-up

Groups share results

Some groups get through both of the examples in 15 minutes; others only do one. It is important to have two groups present their results - one group for the standing wave case and one group for the traveling wave case. Try to leave pictures of the two cases on the board for comparison purposes. Examples a particular group did not do in class should be studied at home.

Compare and contrast

The instructor should encourage students to compare and contrast the results for the two situations. This should include careful attention to:

  1. the energy density distribution (total, kinetic and potential)
  2. propagation of energy density with time.
Animation

The instructor may have a master computer set-up with animation capability. The kinetic energy density, potential energy density and total energy density distributions for both standing wave and traveling wave cases can be plotted and animated in Mathematica (or equivalent) for all to see and discuss. Alternatively, groups that finish quickly can create their own animations in Mathematica or equivalent.

Extensions