18. Fourier Series.


So far we have been working only with sine and cosine waves. Real waves may have very complicated shapes which don't resemble a sine wave. However the French Mathematician Jean Baptiste Joseph Fourier showed that any periodic function can be formed from an infinite sum of sines and cosines. This is very convenient because it means that everything we know about sines and cosines applies to a periodic function of any shape. Although the sum is infinite in theory in many cases using just a few terms may be close enough to provide a good approximation.

This simulation shows the sum of up to 10 harmonics of a sine wave. A harmonic of a sine wave is a sine wave that has a frequency which is a whole number multiple of the frequency of the original wave. The first harmonic (set with the slider H1) is called the fundamental. For standing waves on a string fixed at each end (such as a guitar string) each harmonic is also called a normal mode of vibration. If your computer has speakers you can also hear the wave you see in the graph as an audible sound.

 













Note: Animations may take a few seconds to load.

Questions:

18.1. Move the slider H1 over so that the amplitude of the fundamental is about 1.0. The amplitude shows up in a box to the right of the slider. You can also see the amplitude by holding the mouse the mouse button down and moving the mouse to the top of one of the peaks on the graph on the left. Using the mouse to find the time between peaks, what is the period of this wave? What is its frequency?

18.2. If your computer has speakers try listening to different values on the slider H1 (with all the other sliders on zero). What difference does the amplitude of H1 make in the sound?

18.3. Use the 'clear all' button and move slider H2 (the second harmonic) over to about 1.0. What are the period and frequency of H2? How does this frequency compare with the frequency of H1?

18.4. If your computer has speakers listen to the previous case. How does the sound compare between the wave in 18.1 (using only H1) and the sound in 18.3 (using only H2) when the amplitudes are the same? What numerical quantity determines the pitch of a sound?

You may have noticed a bar in the second graph (to the right of the main graph). This graph is called a Fourier analysis and shows how much of each harmonic is present in the graph on the left. Fourier series usually include sine and cosine functions and can represent periodic functions in time or space or both. In this simulation we only have combinations of sine waves as functions of time. So the Fourier series for the function y(t) showing in the left graph is given by . Here t is time, n is the number of the harmonic or mode (n=1 for the fundamental, 2 for H2 etc.), An is the amplitude of harmonic or mode number n (not the amplitude of the wave itself!) and f is the fundamental frequency (f =1/T).

18.5. Clear the graph and set a fundamental (H1) of amplitude 1.0 and a second harmonic (H2) of amplitude 0.5. Hold the mouse down over each of the bars in the second graph to find the amplitude (second number in the yellow box) and the harmonic (first number in the yellow box) of the combined wave. What are the values of A1 and A2 in the Fourier series of this combination?

18.6. To get the exact shape of an arbitrary periodic function we would need an infinite number of terms in the Fourier series but in this simulation we can only add a maximum of 10 terms. Try the following combination of harmonics and report the approximate shape of the wave you see (you can type the amplitudes into the boxes next the sliders to get exact values): H1=1.0, H2=0, H3=0.333 (=1/3), H4=0, H5=0.20 (=1/5), H6=0, H7=0.143 (=1/7), H8=0, H9=0.11 (=1/9), H10=0. If you have speakers available, play this sound.

18.7. Clear the previous example an try the following combination of harmonics and report the approximate shape of the wave you see (you can type the amplitudes into the boxes next the sliders to get exact values): H1=1.0, H2=-0.5, H3=0.333, H4=-0.25, H5=0.20, H6=-0.166, H7=0.143, etc. If you have speakers available, play this sound.

If you were able to listen to the two waves you constructed in the last two questions you may have noticed they sound different, even though the fundamental (H1) was the same frequency for each case. In other words, although the fundamental pitch is the same (because the fundamental frequency is the same), the two notes sound different because of the amounts and types of harmonics.

18.8. Suppose a clarinet and a trumpet both play the same note (have the same fundamental frequency). Why is it that you can still tell them apart, even though they are playing the same note?

18.9. If you have speakers, try listening to the following combinations. Which sounds more like a trumpet and which sounds more like a clarinet?

18.10. Suppose you wanted to build an electronic instrument which added waves together to imitate other instruments (this is how a musical synthesizer works). What would you need to know about the sound a trumpet makes in order to reconstruct that sound? (Hint: think about the information contained in the graph on the right.)

Credits.
Go To: IUS Physics Top Page.
Contact Dr. K. Forinash, for comments/suggestions/corrections.