The Fourier transform

In-class group activity 5

Find the Fourier transform of a wave pulse

Consider a wave pulse y(x, t) at t = 0. If we want to build this wave pulse by superimposing harmonic waves, we need waves with many different wave numbers k = 2π/λ or wavelength λ. Excel has a Fourier transform function. Given a pulse y(x), Excel calculates the amplitudes of the sine and cosine waves that are needed to synthesize this pulse. These amplitudes as a function wave numbers k = 2π/λ are the magnitude of the Fourier transform of the pulse y(x).

Click here to examine the plots of four wave pulses and the magnitudes of the corresponding Fourier transforms calculated using Excel.

Estimate the width ∆x of the pulse envelop by measuring its full width at half maximum. (The pulse width ∆x has units of mm.)
Estimate the width ∆k of the Fourier transform by measuring its full width at half maximum. (The width of the Fourier transform is measured in units of 1/mm.)
Estimate the product ∆x ∆k. (The product has no units.)

pulse	∆x (m)	∆k (1/m)	product ∆x ∆k
1
2
3
4

What have we learned?