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From this type of observation, a whole slew of theory and algorithms have been developed to characterize such behavior. By taking the Fourier Transform of our sound file, we can see what frequencies make up the signal. Another way of thinking about it, the sound file can be decomposed as the sum of waves of differing frequencies, and the Fourier Transform provides a way of seeing the sound file in the time domain, as in Figures 2 and 3, to the frequency domain shown below in Figure 4.

Spectrogram of figure 1's sound file

Spectrogram (Frequency Domain) plot of the sound file corresponding to plucking of the guitar A string.

The spectrogram shows us how the signal can be broken down into an infinite sum of waves with different frequencies. As Figure 4 shows, the most dominant frequency occurs at about 220Hz. This corresponds to the second harmonic of the signal. The fundamental frequency being 110Hz, which is correct, since the 'A' string of the guitar should be tuned to 110Hz. Apparently the guitar was in tune when the recording was made. Furthermore, you will also notice other dominant waves with frequencies of about 330Hz and 440Hz. Of the musicians out there reading this, this makes sense since 330Hz corresponds to an 'E' which would be the fifth of the 'A' and the third harmonic to the 'A' at 110Hz. But the take away message from these figures is that there are two ways of seeing the same phenomenon of the recorded sound file. One is to simply plot the resulting signal versus time. Another is to view the same signal and its spectrogram, or rather, what frequency components make up the sound file.

Plucking the 'e' string of a guitar

Flash animation of the following figures

Similar to plucking the guitar's 'A' string, here we provide the same graphs but for the sound file in Figure 5.

Plot of the sound signal versus time

The plot of the guitar plucked at the 'E' string over time.

Zoomed in graph of the signal versus time

The zoomed-in plot of the guitar plucked at the 'E' string over time.

Let's point out some differences between Figures 2 and 6. Notice that in Figure 2, the signal decays to 0 a couple of seconds after the signal in Figure 6. Also notice that in Figure 7, the spacing between the oscillations of the signal is smaller than those in Figure 3.

Also, again notice that in Figure 7, the zoomed-in graph of the signal versus time, there are recurring patterns in the signal. See if you can convince yourself that the signal can be described as the sum of waves with differing frequencies.

Spectrogram of figure 5's sound file

Spectrogram (Frequency Domain) plot of the sound file corresponding to plucking of the guitar's high E string.

Figure 8 shows the spectrogram of the sound file in Figure 5. The dominant peak occurs at 330Hz, which from our previous discussion is not surprising. The high 'E' string of a guitar should be tuned to 330Hz.

However, there are some differences to note between Figures 4 and 8. The spectrogram in Figure 8 has less prominent peaks. Between 200 and 300Hz, there seems to be a lump of frequencies that our sound wave has. One of the reasons the spectrum is not as "clean" as the one in Figure 4, is because the high E string causes the lower strings on the guitar to vibrate. Thus one sees lumps in the spectrogram below the fundamental frequency of 330Hz as well as the harmonics of the frequencies of the lower strings.

Extra readings

For more information on the algorithm/procedure that takes you to/from the frequency domain, Derivation of the Fourier Transform and The Fast Fourier Transform (FFT) are good mathematical sources.

Questions & Answers

How we are making nano material?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
what is Nano technology ?
Bob Reply
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The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Stoney Reply
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
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Source:  OpenStax, Music, waves, physics. OpenStax CNX. Mar 15, 2006 Download for free at http://cnx.org/content/col10341/1.1
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