FOURIER ANALYSIS

Is essentially a mathematical prism that allows us to decompose a complex waveform into its constituent pure frequencies, much as a prism separates sunlight into beams of pure color or wavelength. We have just studied the motion of coupled air carts, which gave us an explicit example of a relatively complex waveform to analyze. While the two carts can oscillate with simple sinusoidal motion in the vibrational and sloshing modes of Figures (1) and (2), in general we get complex patterns like those in Figures (3) and (4). What we will see is that, by using Fourier analysis, the waveforms in Figures (3) and (4) are not so complex after all.

Figure 1: Vibrational mode of oscillation of the coupled aircarts. The voltage signal is proportional to the velocity of the cart that has the coil on top.

Figure 2: A pure sloshing mode is harder to get. Here we came close, but it is not quite a pure sine wave.
The MacScope program was designed to make it easy to perform Fourier analysis on experimental data. The MacScope tutorial gives you considerable practice using MacScope for Fourier analysis. What we will do here is discuss a few examples to see how the program, and how Fourier analysis works. We will then apply Fourier analysis to the curves of Figures (3) and (4) to see what we can learn. But first we will see how MacScope handles the analysis of more standard curves like a sine wave or square wave.

Figure 3: Complex motion of the coupled air carts.

Figure 4: Another example of the complex motion of the coupled air carts.

مواضيع ذات صلة

Nonlinear responses

The energy theorem

Quality and consonance

The Fourier series

Musical tones

The speed of sound

The propagation of sound

Resonance in nature

Electrical resonance

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