Coupled Oscillator Chain

# 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(m*x/2) as the initial displacement where m is an integer.

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)

### Credits:

The Oscillator Chain JavaScript Model was developed by Wolfgang Christian using version 5 of the Easy Java Simulations (EJS 5) modeling tool.  Although EJS is a Java program, EJS 5 creates stand alone JavaScript programs that run in almost any browser.  Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP comPADRE collection <http://www.compadre.org/OSP/>.

Coupled Oscillator Model

# Oscillator Chain Model

Let yi= y(xi,t) represent the time-dependent displacement of a particle of mass M with horizontal position xi.  Each particle is coupled to its nearest neighbors with a spring in order to form a chain of oscillators.  It is assumed that the particles move only in the y-direction and that the force Fi on the i-th particle depends on the relative displacement between that particle and its nearest neighbors. The force on the i-th particle can be written as

where K is the Hook's law coupling constant. Because the first and last particles in the lattice are fixed, we compute particle accelerations starting with the second particle and continuing through the lattice until we reach the next-to-last particle. This, the particles in a chain with N oscillators are labeled [0, 1, 2, .... N-1, N, N+1].

One way of understanding a chain of N coupled oscillators of length L and mass M is to study the motion of its normal modes. A normal mode is a special configuration (state) where every particle moves sinusoidally with the same angular frequency ωm. The m-th mode Φm of the oscillator chain of length L is

.

The system stays in a single mode and every particle oscillates with constant angular frequency ωm if the oscillator chain is initialized in a single mode.

An arbitrary initial configuration can be expressed as sum of these normal modes.

### Software Requirements

SoftwareRequirements

 Android iOS Windows MacOS with best with Chrome Chrome Chrome Chrome support fullscreen? Yes. Chrome/Opera No. Firefox/ Sumsung Internet Not yet Yes Yes cannot work on some mobile browser that don't understand JavaScript such as..... cannot work on Internet Explorer 9 and below

### Credits

Wolfgang Christian; This email address is being protected from spambots. You need JavaScript enabled to view it.

### Pictures

Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian

 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, with options created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577
 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577

 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577

 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577

 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577

 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577

 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577

 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577

 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577

 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577

 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577

 Coupled Oscillator Chain Classical Wave Function by Professor Wolfgang Christian with ideas from Francisco Esquembre and Dieter Roess, created by Loo Kang WEEhttp://iwant2study.org/ospsg/index.php/577

### Coupled Oscillator Chain by Professor Wolfgang Christian

"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 Easy JavaScript Simulation (Ejss) model shown here contains 100 31 coupled oscillators equally spaced within the interval [0, 2π] with fixed ends. The m-th normal mode of this system can be observed by entering f(x) = sin(m*x/2) as the initial displacement where m is an integer. For simple layman, options were created associate mathematical initial conditions to actual real life physics, such as stationary waves, exponential and triangle wave pulse etc.

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.

### Theory by Professor Wolfgang Christian

Let yi= y(xi,t) represent the time-dependent displacement of a particle of mass M with horizontal position xi.  Each particle is coupled to its nearest neighbors with a spring in order to form a chain of oscillators.  It is assumed that the particles move only in the y-direction and that the force Fi on the i-th particle depends on the relative displacement between that particle and its nearest neighbors. The force on the i-th particle can be written as

$$F_{i} = -K[ y_{i+1}-y_{i} - ( y_{i}-y_{i-1}])$$

where K is the Hook's law coupling constant. Because the first and last particles in the lattice are fixed, we compute particle accelerations starting with the second particle and continuing through the lattice until we reach the next-to-last particle. This, the particles in a chain with N oscillators are labeled [0, 1, 2, .... N-1, N, N+1].

One way of understanding a chain of N coupled oscillators of length L and mass M is to study the motion of its normal modes. A normal mode is a special configuration (state) where every particle moves sinusoidally with the same angular frequency ωm. The m-th mode Φm of the oscillator chain of length L is

$$\Phi _{m} (x,t) = sin (\frac{m \pi x}{L}) cos ( \omega _{m} t + \phi )$$

.

The system stays in a single mode and every particle oscillates with constant angular frequency ωm if the oscillator chain is initialized in a single mode.

$$\omega ^{2} _{m} = \frac{4K}{M}sin^{2}(\frac{m\pi}{2N})$$

An arbitrary initial configuration can be expressed as sum of these normal modes.

### 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.

1. "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)
2. "One-dimensional lattice dynamics with periodic boundary conditions: An analog demonstration," Jon H. Eggert, Am. J. Phys. 65, 108 (1997)
3. "Evolution of a vibrational wave packet on a disordered chain," Philip B. Allen and Jonathan Kelner, Am. J. Phys. 66, 497 (1998)

### Credits:

The Oscillator Chain JavaScript Model was developed by Wolfgang Christian using version 5 of the Easy Java Simulations (EJS 5) modeling tool. Although EJS is a Java program, EJS 5 creates stand alone JavaScript programs that run in almost any browser. Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP comPADRE collection <http://www.compadre.org/OSP/> and the remixed version on Open Source Physics Singapore collection <http://iwant2study.org/ospsg/>.

### Other Resources

1. https://www.compadre.org/osp/items/detail.cfm?ID=12977 Oscillator Chain JS Model written by Wolfgang Christian
2. https://www.compadre.org/osp/items/detail.cfm?ID=11523 Vibrating String PDE Model written by Francisco Esquembre and Dieter Roess
3. http://physics.bu.edu/~duffy/HTML5/transverse_standing_wave.html