# Coupled Oscillator Chain

"The investigation by John and Daniel Bernoulli [of the coupled oscillator chain] may be said to form the beginning of theoretical physics as distinct from mechanics, in the sense that it is the first attempt to formulate the laws of motion of a system of particles rather than that of a single particle." Leon Brillouin

Oscillator Chain models a one-dimensional linear array of coupled harmonic oscillators with fixed ends. This model can be used to study the propagation of waves in a continuous medium and the vibrational modes of a crystalline lattice. The Ejs model shown here contains 31 coupled oscillators equally spaced within the interval [0, 2 π] with fixed ends. The m-th normal mode of this system can be observed by entering f(x) = sin( mx/2) as the initial displacement.

Wave propagation can be studied by entering a localized pulse or by setting the initial displacement to zero and dragging oscillators to form a wave packet. An interesting and important feature of the Oscillator Chain model is that the speed of a sinusoidal wave along the oscillator array depends on its wavelength. This causes a wave packet to disperse (change shape) and imposes a maximum frequency of oscillation (cutoff frequency) as is observed in actual crystals.

## References:

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

- Analytical Mechanics 5 ed by Grant R. Fowles and George L. Cassiday, Saunders College Publishing (1993)

There are many laboratory and computer experiments that build on the basic 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 lattice dynamics with periodic boundary conditions: An analog demonstration,"
  Jon H. Eggert, Am. J. Phys. 65, 108 (1997)
- "Evolution of a vibrational wave packet on a disordered chain," Philip B. Allen and Jonathan Kelner,
  Am. J. Phys. 66, 497 (1998)

### 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/>.