# Building Intuition for Fourier Series/Transforms

Initial Publication Date: September 21, 2022

## Summary

This activity consists of two parts:

1. Represent functions using Fourier series.
2. Understanding how a function's properties manifest in reciprocal space.

Used this activity? Share your experiences and modifications

## Learning Goals

During the first activity, the students learn how a series of sines and cosines can be used to represent a function. They also learn that a finite number of terms is sufficient to represent a function. In that context, the Fourier coefficients here are a nice segue into Fourier transforms and reciprocal space. As a bonus, the Gibbs phenomenon is visible and can be discussed in class.

For the second activity, students explore how real space and Fourier space are related. The idea of reciprocal space is introduced.

## Context for Use

The educational leveel for this class are undergraduate physics students at a liberal arts institution. This is done in classroom during a time frame of only 10-15 minutes. The students need very little technical expertise to download, open and run the app; however, some students are familiar with MatLab and they get curious about how to make an app like this for projects later in the semester.

The beauty of this activity consists of the fact that we do this activity BEFORE the math for Fourier transforms. They are somewhat familiar with the idea of recripocal space from diffraction. It is a great activity to do before or after introducing the math for Fourier transforms in the context of diffraction, electric postential, and Fourier analysis in quantum mechanics.

## Description and Teaching Materials

Download the Matlab apps named FSeries.mlapp and FTransform.mlapp. Open the FSeries.mlapp activity first. Move the slider back and forth to observe how many terms are needed to obtain a square wave. How many terms would be enough? What are the periods of the sinusoidal waves needed to build the square wave?

Open the FTransform.mlapp activity second. Move the slider back and forth to observe the relationship between the top-hat function and the Fourier transform. What do you observe about the relationship betweeen the widths of both patterns.

Fourier Series ( 48kB Sep21 22)
Fourier Transform of a Top-Hat Function ( 71kB Sep21 22)

## Teaching Notes and Tips

This is a great exercise to introduce, remind, and reinforce Fourier series and transforms. As a next step, let the students build their own app.

## Assessment

There are brief quiz multiple choice questions on Fourier transforms and reciprocal space. For example, if the width of the top-hat function widens, does the Fourier transform (a) widen (b) becomes more narrow, (c) remain unchanged.

## References and Resources

This is a brief video on how to package MatLab apps https://www.youtube.com/watch?v=RGUOpZyRvgA

You need to download and install the Matlab compiler in order to be able to package apps.