Physics II Activity Guide.  Spectral Analysis of Sound
Sound (Ref: Hecht ch 11, especially “Superposition” and “Fourier Analysis”, pp 464-467)

Note for 2003-04.  This activity needs some editing to make more well-defined experiments.
 

Objectives and Goals
Use the computer (Graphical Analysis) to synthesize complex waveforms by means of the addition of harmonics.
Work with the idea of expressing waves as spectra as well as displacement vs. time.
Analyze various sounds from the study of the time-dependent waveforms and by analyzing the spectra of the various waveforms.
I. What do you think? Consider a pure tone such as
y = y0sin(w t).

a) Sketch the wave in your notes.

b) If you were to analyze the spectrum of this wave (a plot of amplitude vs frequency) sketch what you think the spectrum looks like. Be sure to label the axes.

c) Suppose you have a "sawtooth" wave as indicated by the instructor. Sketch both the wave and its spectrum that you expect.
 

Observe sounds with a data acquisition system. Use the Logger Pro configuration for recording sound waves and their spectra (sound_level.mbl)
  • Tuning fork emitting pure tone. A tuning fork when struck by a soft object, such as a person’s knee or a soft mallet, should produce a pure tone or a simple sine wave. Record the wave and the spectrum of the tuning fork.
  • Tuning fork pure tone at different pitch. Compare the waveform and the spectra.
  • Tuning fork emitting a “ding” sound. This can be obtained by hitting the tines of the tuning fork on a hard object such as a desk-top or someone’s skull.
  • Try to sing a pure tone. This is best achieved by singing “oooooh”. Males use a falsetto voice.
  • Sing an “ahhhh” which has some structure in the sound wave. Sketch the spectrum.
  • Sing a very nasal “eeeee”.
  • Make sure that the waveforms are not “cut-off” at top or bottom. If so, you must compensate by less sound intensity into the microphone. If necessary, re-scale the vertical axes of the graphs so that the waveform is centered.
    • All these tones should by sung to the same pitch. You should practice your singing so that you see a “good” waveform on the oscilloscope Use a tone generator to keep your pitch constant.
    Question: Describe the basic difference in the shapes for the different sounds.
    III. Save waveforms for later analysis. Obtain a good waveform for each of the sounds above, all sung to the same pitch.. You may use the hard drive and an appropriate directory (amlab or pmlab).
    Musical instrument. Record additional waveforms for the following:
    d. Musical instrument playing low pitch
    e. Same instrument playing high pitch
    f. Noise such as "shhhh"
    Fourier Theorem.  State Fourier Theorem.  Explain how your waveforms and sound spectra support Fourier Theorem.

    IV.  Synthesize Complex waveforms with computer.We will synthesize a complex wave by adding sine waves of several frequencies according to:
     
     

    ytot = sin(wot) + (1/2) sin(2wot) + (1/3) sin(3wot) + ...

    (1)
    Use the spreadsheet (Graphical analysis) to produce each of the terms in Eq. 1 above. Call each term y1, y2, and y3 respectively for the “harmonic” series with a fundamental frequency of 20 Hz.
     
    The last plot contains a wave that looks something like a sawtooth wave. Note that the complex wave contains the fundamental at a frequency of 20 Hz. The 2nd wave is 2 times the fundamental frequency and the third wave, y3, is 3 times the fundamental frequency. The complex wave contains all three frequencies.

     
     

    Spectral analysis of Simple Waves.

    The spectrum of a wave is a graph in which the horizontal axis is frequency and the vertical axis is amplitude as opposed to a time-series graph (signal vs. time) that an oscilloscope displays.
     
     

      Consider a pure tone, y1, which is a sine wave at a frequency of 20 Hz. Predict the spectrum of this signal: a graph of amplitude vs. frequency. This is tricky. There is only one frequency present.

      Analyze the spectrum of the pure tone, y1. The tool in Graphical Analysis is FFT for Fast Fourier Transform. Obtain this tool by:

      Click on the graph window to select the time-series graph.

      Click on “Window” from the menu bar and create a New window of type FFT graph. It is most convenient if the window layout for the new window is just beneath the time-series graph.

      Choose the appropriate columns for the FFT setup. The number of points in the FFT should most closely match the number of points in your waveform (512 for this exercise).

      Explain the meaning of the FFT graph and compare with your prediction.

      Predict the spectrum of the pure tone y2 which is at twice the frequency of y1.

      Do it. Click on the frequency axis to change the FFT setup. Explain the result and compare with your prediction.

      Predict the spectrum for the complex synthesized waveform (y1+y2+y3).

      Do it and explain.

      Analyze pre-digitized sounds from part I.
      Open ended: Examine more waveforms and spectra, including other instruments, voices, pitches and include in your summaries.

      Detailed Analysis For one of the repeating complex waveforms, and its spectrum compare the lowest frequency of the spectrum with the period of the complex waveform from the time-series graph. The lowest frequency should equal the fundamental and the separation of the frequency peaks should also equal the fundamental (as in the synthesized spectrum of part IV.

      Analyze a "Pure Sawtooth" Wave.

      Load Graphical Analysis and open a file called "Sawtooth.dat". The path to this file is Classes 'Raptor'(G:)/ physics/Sawtooth/Sawtooth.dat. You will observe a pure sawtooth wavform that does not have any of the rounded features seen in the synthesized waveform. The pure sawtooth waveform is the result of an infinite Fourier series.
      Compare the spectrum of the pure sawtooth with that of the  wave you synthesized.
    Your summary should contain a discussion for all parts I - V of this handout. You should show that you understand the concept of sound spectra and the relationship of spectra and the waveforms.

     
     

    Problem Solving

    Problems will be assigned from the text.