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Energy Density of Waves on a String

Keywords: Oscillations & Waves, Middle Division, Transfer of Energy in Waves, Transverse Pulses and Waves, Standing Waves, Energy Density

Highlights of the activity

This small group activity is designed to help upper-division undergraduate students understand solutions of the non-dispersive wave equation, kinetic and potential energy densities, and energy propagation and conservation.
Students determined the energy density distribution for standing and traveling waves on a string.
The compare and contrast wrap-up discussion focuses on the energy distribution and propagation of energy density with time.

Reasons to spend class time on the activity

This activity provides an opportunity for students to consider the energy propagation and energy density of a wave on a string. In their lower division courses, students typically perform an energy analysis of a mass-spring system. Middle division students inappropriate rely on their intuition of spring-mass systems while doing an energy analysis of a wave on a string (many students want to consider each section of rope as an isolated system in which energy is conserved). This activity allows students to use many types of representations to make sense of energy of a wave, especially in connecting an algebraic description of the energy (in terms of time and space derivatives of the wave function) to pictures of the displacement of the wave and to physical intuition about the speed and degree of stretch of different portions of the string. Students also compare the energy propagation in the cases of standing and traveling waves.