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Find the Fourier transform of a plane wave.
If students know about the Dirac delta function and its exponential representation, this is a great second example of the Fourier transform that students can work out in-class for themselves.
Students will need a short lecture giving the definition of the Fourier Transform \begin{equation} {\cal{F}}(f) =\tilde{f} (k)= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}\, f(x)\, dx \end{equation}
Students may ask what is meant by a plane wave. Help them figure out what is meant, from the context or give them the formula if time is tight.
Keep the time dependence in or leave it out depending on how much time you have to deal with a little extra algebraic confusion.
group Small Group Activity
5 min.
assignment Homework
format_list_numbered Sequence
accessibility_new Kinesthetic
30 min.
group Small Group Activity
30 min.
group Small Group Activity
60 min.
Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.
Each group is given a different two-dimensional vector \(\vec{k}\) and is asked to calculate the value of \(\vec{k} \cdot \vec {r}\) for each point on the grid and to draw the set of points with constant value of \(\vec{k} \cdot \vec{r}\) using rainbow colors to indicate increasing value.
assignment Homework