Student Guide: Making Waves by Visualizing Fourier Transformation

Abstract

Using a simple graphical presentation we can visualize the integrand of the forward and inverse Fourier transforms as a topographic surface. This presentation aids in understanding frequency domain and real-space relationships, such as the important, but often poorly understood, contributiaon of the frequency domain phase spectrum to the real-space shape. The Fourier transform integrand visualization method presented here can also help develop insights into complex wave behavior, such as the relationship between traveling and standing waves and the evolution of dispersing wavetrains. Fourier analysis is often introduced with a figure showing how to approximate a function by adding together sinusoids (Figure 1A). I extend this presentation through the introduction of Fourier transform integrand visualization (FTIV) and use this technique to illustrate the relationship between traveling and standing waves, the Fourier shift theorem, and how the phase affects the real-space shape of a dispersing wave. Figure 1(A) shows a real space, in this case a time-domain function, a boxcar symmetric about t = 0, and the first few terms of its Fourier series \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[ u(t)=\frac{c\_{0}}{2}+\begin{array}{c}{\infty}\\{{\sum}}\\n=1\end{array}[c\_{n}\mathrm{cos}({\omega}\_{n}t+{\phi}\_{n})],{\omega}\_{n}=n{\omega}\_{1},a_{n}{ }\mathrm{real}.\] \end{document}(1) The weights, c ( ωn ), and phase shifts, ( ϕn ), represent u in the frequency domain. The cos ( ωnt ) terms are known as basis functions and have variables from both domains, i.e. , time and frequency, in their argument. The set of basis functions must meet two important conditions. First, mutual orthogonality, which means you cannot make any of them by summing the others. Second, “completeness,” which means any arbitrary function, with some conditions to ensure convergence, can be represented using only this set of functions. The Fourier series is also periodic, with period T = 2 π/ω 1. The Fourier series can be generalized to the inverse Fourier transform \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[ u(t)={{\int}}_{-{\infty}}^{{\infty}}c({\omega})\mathrm{cos}({\omega}t+{\phi}({\omega}))d{\omega}\] \end{document}(2) where the amplitude, c ( ω ), and phase, ϕ ( ω ), spectra come from the …