Presenting these definitions in between students Emulating a Wave in a Periodic System is highly recommended. Doing so will help solidify the connection between the verbal and physical representations.

1. 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.)

1. 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.
2. Dispersion Relations:
   • Relate the shape of the envelope function to the normal mode frequency.