pISSN 0374-4914 eISSN 2289-0041

## Research Paper

New Phys.: Sae Mulli 2020; 70: 880-884

Published online October 30, 2020 https://doi.org/10.3938/NPSM.70.880

## Analytic Solutions for Vibrating Strings with Varying Mass Density

Won Sik L'YI

Department of Physics, Chungbuk National University, Cheongju 28644, Korea

Correspondence to:wslyi@chungbuk.ac.kr

Received: June 29, 2020; Revised: July 14, 2020; Accepted: July 15, 2020

### Abstract

In general, the analytic solutions of one-dimensional wave equations with position-dependent mass densities cannot be found. In this study, we solve the wave equations of strings for the cases of linearly or exponentially changing mass densities. When the mass density changes linearly, the solution can be written in terms of the Bessel function $J_{1/3}$ and the Neunmann function $N_{1/3}.$ For large position $x,$ the spatial form of the wave changes as $1/\sqrt[4]{x},$ and the spatial oscillation is narrowed with the length which is proportional to $1/\sqrt{x}.$ When the mass density changes exponentially, the solution can be written in terms of the Bessel function and the Neunmann function with an imaginary order. As $x\to -\infty,$ the solutions reduce to sine and cosine functions. However, as $x\to \infty,$ two types of solutions exist, one increasing rapidly, and the other decaying rapidly. For a string of finite length both solutions are possible, but when a string stretches to $x\to\infty,$ the divergent solution is not allowed, and the string has fixed nodes.

Keywords: String vibration, Varying mass density, Analytic solutions