Notes on & illustrations of basic language of functions that are harmonically varying in space basic_wavefunctions_space_wiki.ppt

The students are thoroughly familiar with descriptions of quantities that oscillate in time \[\psi \left( t \right)=A\sin \left( \omega t+\varphi \right)\] Introduce the corresponding spatial analogs:

• wavelength, lambda, as the repeating distance in space, similar to period, $T$, the repeating unit in time.
• $k$, the “wave vector”, = $\frac{2\pi}{\lambda}$, with the same relationship to $k$ as angular frequency, omega, has to $T$.
• sometimes (not in this course) use “wave number”, $\bar{\lambda}= \frac{1}{k}$, analogous to $f = \frac{1}{T}$.
• phase constant, $\varphi$, determines origin in space, similar to oscillations in time.
• Remind students of 4 equivalent ways to represent a real, sinusoidally varying quantity:

\[\psi\left( x \right)=A\sin \left( k x+\varphi \right) \qquad\hbox{(“A-form”),} \]

\[\psi\left( x \right)=B_{p}\cos k x+B_{q}\sin k x \qquad\hbox{(“B-form”),} \]

and the two complex-number-containing forms,

\[\psi \left( x \right)=Ce^{i k x}+C^{*}e^{-i k x} \qquad\hbox{(“C-form”),} \]

\[\psi\left( x \right)=\Re\left( De^{i k x} \right) \qquad\hbox{(“D-form”).} \]

Use something similar to the “wave machine” depicted here to show the students a wave-like disturbance. Computer animations work, too, but the hand-cranked piece of machinery strikes a chord! Any other physical example that clearly shows wavelength and propagation would be good.

A review of the relationships among the coefficients is in order, usually as a short group activity in which various groups are assigned different pairs of the above.

picture of wave machine