Superposition of Sine Waves
Excursion into physics
Sound waves can be mathematically represented as a combination of sine waves. Every musical tone is composed of several sine waves of form y(t) = a sine(ω t + φ).
The amplitude a influences the volume of the tone while the angular frequency ω determines the pitch of the tone. The parameter φ is called "phase" and indicates if the sound wave is shifted in time.
If two sine waves interfere, superposition occurs. This means that the sine waves amplify or diminish each other. We can simulate this phenomenon with GeoGebra in order to examine special cases that also occur in nature.
Visualization of the superposition of sine waves
Instructions
| 1. |  | Create three sliders a_1, ω_1 and φ_1 using the default settings for sliders. |
Hints: The input _1 produces an index 1.
In order to insert a Greek letter, place the cursor in the Name text field and click on the appearing letter on the right side of the text field. This opens a list of Greek letters to choose from.
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| 2. |  | Enter the sine function g(x)= a_1 sin(ω_1 x + φ_1). |
| 3. | | Create three sliders a_2, ω_2 and φ_2, again using the default settings for sliders. Hint: Sliders can be moved in the Graphics View when the Slider tool is activated. |
| 4. | | Enter another sine function h(x)= a_2 sin(ω_2 x + φ_2). |
| 5. | | Create the sum of both functions sum(x) = g(x) + h(x). |
| 6. |
| Use the Style Bar in order to change the color of the three functions and their corresponding sliders so they are easier to identify. |
Try it yourself...
Back to school...
Examine the impact of the parameters on the graph of the sine functions by changing the values of the sliders. Set a1 = 1, ω1 = 1 and φ1 = 0.
- For which values of a2, ω2 and φ2 does the sum have maximal amplitude? Note: In this case the resulting tone has the maximal volume.
- For which values of a2, ω2, and φ2 do the two functions cancel each other? Note: In this case the tone cannot be heard any more.