Energy in one dimensional linear waves in a string

TL;DR

This paper examines the inaccuracies in textbook formulas for energy in one-dimensional waves on a string, especially for standing waves, and proposes corrections suitable for introductory physics courses.

Contribution

It identifies errors in standard energy expressions for waves and provides corrected formulas tailored for calculus-based physics education.

Findings

Standard textbook formulas can give incorrect energy calculations for certain wave cases.

Corrected expressions are proposed to accurately describe energy in small amplitude waves.

The corrections are suitable for teaching introductory physics courses.

Abstract

We consider the energy density and energy transfer in small amplitude, one-dimensional waves on a string, and find that the common expressions used in textbooks for the introductory physics with calculus course give wrong results for some cases, including standing waves. We discuss the origin of the problem, and how it can be corrected in a way appropriate for the introductory calculus based physics course.