# Traveling Waves on an Oscillator Chain

Driven Oscillator Chain models a one-dimensional linear array of coupled simple harmonic oscillators driven at one end and attached to a sliding rod shock absorber at the other. The mass and damping of the shock absorber are chosen to eliminate reflections at the driving frequency.

A traveling wave propagates along the oscillator chain in the positive x-direction with a wave function Φ(x,t) that depends on position and time

where A is the amplitude, ω=2πf is the angular frequency, and k=2π/λ is the wave number. The angular frequency and wave number are related though the dispersion relationship

.

An interesting and important feature of this dispersion relationship is that it predicts a maximum frequency wH=(4K/M)1/2. What type of motion is observed if the diving frequency is greater than this maximum?

## References:

The coupled oscillator (beaded string) model is discussed in most intermediate mechanics textbooks.

- Analytical Mechanics 5 ed by Grant R. Fowles and George L. Cassiday, Saunders College Publishing (1993)
  Section
- Vibration and Wave Physics 3 ed by Iain G. Main, Cambridge (1993). See Chapter 9 for a discussion of non-dispersive traveling waves and impedance matching.

There are many laboratory and computer experiments that build on the beaded string model.

- "Normal modes and dispersion relations in a beaded string: An experiment for an undergraduate laboratory,"
  Gauri Shanker, V. K. Gupta, N. K. Sharma, and D. P. Khandelwal, Am. J. Phys. 53, 479 (1985)
- "One-dimensional Wave Demonstration," John M. Fowler and John T. Brooks, Am. J. Phys. 35, 1065 (1967)

### Note:

This simulation was created by Wolfgang Christian using the Easy Java Simulations (Ejs) modeling tool. You can modify this simulation if you have Ejs installed by right-clicking within a plot and selecting "Open Ejs Model" from the pop-up menu. Information about Ejs is available at: <http://www.um.es/fem/Ejs/>.