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=====Approximating the N-th Normal Mode Frequency for an N-chain Oscillator (10 minutes)=====
  * If an N-chain oscillator is oscillating at the N-th normal mode in the first Burilloun zone (that is, when the wavelength of the envelope function is $k=\frac{\pi}{a}$), the frequency of oscillation can be reasonably approximated using the equation of motion for a particle in the system.


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  * The key approximation for this calculation is that the displacement of each molecule in the oscillator is equal in magnitude.  Using this approximation, the equation of motion for a particular particle becomes

$$m\ddot{x}=-2\kappa x \; - \; 2\kappa x \; \; . $$

Assuming that the equation describing the particle's motion has the form

$$x(t)=Ae^{i \omega t} \; \; , $$

this equation can be inserted into the equation of motion to find that

$$\omega=\sqrt{\frac{4\kappa}{m}} \; \; . $$

//Have the students test this approximation using the "One Dimensional Oscillator Chain" program.  This exercise works quite well as an extension in concluding the [[..:..:..:courses:activities:ppact:ppperiodiclab1|Coupled Oscillators and the Monatomic Chain Lab]].//

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