The students are shown the graph of a function that is a superposition of three harmonic functions and asked to guess the harmonic terms of the Fourier series. Students then use prewritten Sage code to verify the coefficients from their guess. The program allows the students to enter functions of their own choice as well as the one that is preset.
- Go to Fourier Basis Functions and play with the simulation near the top of the page.
- What function does \(a_m\) correspond to? What does \(m\) mean?
- What function does \(b_m\) correspond to? What does \(m\) mean?
- What values can \(m\) take?
- Go to Fourier Series: Exploration and look at the simulation near the top of the page. It shows the graph of a function in blue.
- Move the sliders until the green curve matches the blue one. Only three of the sliders need to be set to nonzero values.
- Based on the graph, why might you anticipate which values of \(a_m\) and \(b_m\) are nonzero, larger or smaller, positive or negative?
First, make sure that students understand what it means to add two functions (pointwise). The SWBQ: Adding Functions Pointwise can be helpful with this.
Students can also be asked to build any unique superposition function using the PhET simulation Fourier: Making Waves. This helps the students to grasp and apply the idea of superposition.