An Algebraic Derivation of the Standing Wave Problem

TL;DR

This paper presents an algebraic derivation of standing wave solutions in mass-spring systems, revealing how wave shapes and frequencies emerge from algebraic and physical principles.

Contribution

It introduces a novel algebraic approach to derive standing wave solutions and their frequencies in both discrete and continuous mass-spring systems.

Findings

Standing wave shapes are derived algebraically.

Allowed frequencies are determined for discrete and continuous systems.

Wave formation is explained through centripetal force and projection onto the plane.

Abstract

The standing wave solution on an idealized mass spring system can be found using straight forward algebra. The solution is found when this system makes jump rope like rotations around an axis.The standing wave forms a constant shape in a radial direction using the centripetal force condition. The wave is projected back onto the x,y plane to get the planar time dependent solutions. The allowed frequencies are found for a discrete system as well as a continuous system.