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

Main Ideas

  • Solutions of the non-dispersive wave equation
  • Kinetic and potential energy densities
  • Energy propagation and conservation

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.


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: $$\psi(x,t) = Bsin(kx+\phi)cos(\omega t+\delta)$$


  • Table-top whiteboards with markers
  • A handout for each student
  • Front of room blackboard/whiteboard
  • Instructor's Mathematica/Maple worksheet for wrap-up.

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

  • Because they are given a generic algebraic equation for energy density, many students will begin by writing an equation for their wave form. This is a good opportunity for them to practice how to identify standing and traveling waves from the algebraic form.
  • Some students may also draw a picture of the wave at various times to see how the wave form changes. This is another good opportunity for students to match algebraic descriptions of the wave form to pictorial representations.
  • Few students attempt to graph the time and spatial derivatives of the wave form. This is a good opportunity to challenge students to think about their problem in more than one way.
  • Students usually recognize that the potential energy of the string is proportional to the spatial derivative of the waveform, and correctly identify where that derivative is maximal, but only a few appreciate (without prompting) that the string is stretched maximally in this region and therefore the rope should store PE where it is stretched.
  • Students all know that the kinetic energy is maximal where the rope is moving fastest, but don't always know how to decide where this place is. The instructor can guide them through this process of drawing the waveform at two infinitesimally displaced times, and once they see this device, the students can usually decide where the string is moving fastest. Another option is to have them plot $\frac{\partial \psi}{\partial t}$ as a function of location for some specific time values and see where the rope speed is largest.
  • In considering specific points on the rope, many students will try to make an analogy with a mass on a spring. This analogy tends to falsely lead students to believe that for each point on the rope, the total energy of that point must be conserved (“But I thought energy was constant?”). Try to get students to articulate what energy conservation means.
  • An important aspect of this exercise is for students to understand that energy propagates along the rope in the case of a traveling waves. The students must come to grips with this fact and reconcile this with the notion of energy conservation.
  • Students also confuse distance from equilibrium of the rope with distance from equilibrium of the mass-spring. In both cases, the potential energy is related to stretching - the rope stretching and the spring stretching. The rope is stretched maximally where the slope of the rope is maximum (at the “equilibrium” location), as opposed to the mass-spring case where the maximum potential energy occurs when the mass is at its maximum displacement. Having the students draw the rope as a series of dots that move vertically, with the distance between the dots indicating the stretch of the rope, helps students realize that the rope stretches most where the slope of the rope is greatest.

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.

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.


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