Workout: Spectral analysis

Read

Spectral analysis -- summing different wavelengths

Launch

the PhET simulation Fourier: Making Waves.

Set up

When the program comes up, it is set to show you the result of adding soundwaves produced by the sum of harmonics, like those produced by standing waves on an stretched elastic string with both ends tied down.

Answer these questions

1. When we studied beats, we added two oscillations with nearby frequencies. Now, we'll look at adding waves whose frequencies are multiples of each other. This has an interesting consequence. If the frequencies that are added are multiples of each other what does that mean about the periods? The signal from a single-frequency wave repeats in time intervals of the period. If a signal consists of a sum of signals whose frequencies are multiples of each other, does it ever repeat? If it does, in what period?

2. On the "Discrete" tab, it should show a single sine wave. Set the "Graph controls" to "time" and the shape to "cos" (radio button). Turn on the sound. Now add a second harmonic by entering "0.50" in the slot A2. What happens to the shape of the sum? Explain why it does this in terms of what you see in the shapes being added. What do you think the graph will look like if you go to larger and smaller times than are shown on the graph? Turn on sound. How does adding a harmonic change the sound?

3. Choose the Function "Wave packet." Your screen should look like the figure at the left.  Why does the result look the way it does? Answer in terms of the properties of the functions being added and the shape of the resulting pulse. (If you hover over the bars that show how much of each harmonic is included, it while highlight that function in the Harmonics pane and suppress the rest.) How is the sound different from a simpler tone? 

4. Go to the "Discrete to Continuous" tab and explore the available controls and tools. State one interesting result that you learned from your explorations.

Joe Redish 4/8/18