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=====Definitions of Important Terms for This Unit (10 minutes)=====

Presenting these definitions in between students [[..:..:activities:ppact:ppwaveemulate|Emulating a Wave in a Periodic System]] is highly recommended.  Doing so will help solidify the connection between the verbal and physical representations.

  - Envelope functions:
    * Describe the displacement of discrete atoms in an oscillating system.  That is,if an envelope function is described by $\psi (x,t)$, the value of the function will describe the displacement of a particle at location $x$ and time $t$ .
    * //**To relate this to the wave emulation, have the students perform the activity in front of a blackboard.  Before the students start oscillating, draw the envelope function describing them on the board behind them.  Tell them this is the envelope function describing the particles at time $t=0$.**//



(As a side note, you can also note that an envelope function of wavelength $a$, where $a$ is the separation distance between each molecule, is equivalent to an envelope function of wavelength $\lambda -> \infty $.  Have the students perform this scenario if you wish.)


  - Normal modes:
    * Are a special set of envelope functions.
    * Are special //because// each atom in the system oscillates with the same frequency when that normal mode is excited.
    * Occur from satisfying particular boundary conditions (fixed, periodic, conditional, etc.).
    * //**To relate this to the wave emulation, discuss the choices of $k$ made for each oscillating system.  For example, the instructor can note in some case that the ends of the wave are moving exactly in phase with each other; this satisfies periodic boundary conditions.**//
  - Dispersion Relations:
    * Relate the __shape of the envelope function__ to the normal mode frequency.



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