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...

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