Products of harmonic functions and projections
Highlights of the activity
- This small group activity is designed to help upper-division undergraduate students to understand, graphically, the orthogonality among sinusoidal functions whose frequencies are multiples of a fundamental frequency.
- Students are presented with different pairs of harmonic basis functions and asked to determine graphically whether the integral of the product of the two functions over one period of the fundamental is non-zero. Groups then calculate the integral analytically to compare with the previous task.
- The compare and contrast wrap-up discussion revolves around the concepts of projection and orthogonality, and introduces the Kronecker delta notation.
Reasons to spend class time on the activity
This activity promotes a graphical (or geometric) understanding of what it means to project one function onto another. This process is the essence of Fourier (or any other) decomposition.
The ability to multiply (point-wise) two functions of a variable, presented graphically, is not yet part of the students' toolbox, and it's a useful skill. Identifying zeroes, looking for patterns and symmetries, and generalizing to other cases are all needed to complete this activity.