Lattice Elasticity, the Vibrating String, and Nonlinearity

Developed by W. Freeman

The standard description of the behavior of the vibrating string is only valid in the low-amplitude limit. In this exercise, students build a lattice-elasticity model of a vibrating string and then study its properties, first verifying that the standard properties predicted by v=Tμ hold in the low-amplitude limit and then studying the nonlinear properties of the vibrating string as the amplitude increases.

Subject Areas Waves & Optics, Mathematical/Numerical Methods, and Other
Levels Beyond the First Year and Advanced
Available Implementation C/C++
Learning Objectives

  • Students will be able to construct a discretized model of a continuum system that reproduces the continuum properties, and understand its limits of validity (Exercises 1, 4, and 5)

  • Students will be able to code numerical models involving arrays of dynamical variables (Exercise 2)

  • Students will be able to test a dynamical model using conservation of energy and visualizations to verify its behavior (Exercise 3)

  • Students will become familiar with the low-amplitude and high-amplitude phenomenology of the vibrating string (Exercises 4-6)

  • Students will be able to flexibly analyze data from numerical models to determine a power-law relationship (Exercise 6)

  • Students will become familiar with “perturbative thinking” and the characteristics of the onset of non-ideal behavior (Exercises 5-6).

Time to Complete 90-360 min