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PHYS 302 : Unit 9 Reading Notes – Athabasca University 1 PHYS 302: Vibrations and Waves - Unit 9 Reading Notes French, A. P. Vibrations and Waves . New York: W.W. Norton & Company, 1971. Chapter 6: Normal Modes of Continuous Systems. Fourier Analysis (pp. 170–189, up to “Fourier Analysis”). Please note that the order of materials is different in the textbook than in the video lectures. Chapter 6: Normal Modes of Continuous Systems. Fourier Analysis Longitudinal Vibrations of a Rod (p. 170) In our discussion so far, we have stressed the importance of boundary conditions. To some extent we have explored how to obtain boundary conditions. With strings, we concentrated on fixed ends, which are necessary to keep tension in the string during continuous vibration. The “massless, frictionless string” the video lecture refers to would be able to produce the needed tension on an end that is not fixed, but such a string exists only in theory. Other systems can have other boundary conditions. The lecture focuses on sound cavities, which are covered in the next text section. This section examines the somewhat similar case of vibrations of a rod. In the video lecture, Dr. Lewin demonstrates a free end in the similar case of a torsional vibration. Longitudinal vibrations of a rod share similar properties to that of a massive spring, which we looked at in Chapter 3. They are also similar to torsional oscillations (pp. 54–57). Pages 170–171 refer to Young’s modulus. You can find a table of Young’s moduli Y on page 56. Note that for common solid materials, Y is very large! Even a small deformation returns a lot of force. These stiff (solid) materials support very fast sound speeds (see below) and high frequencies. Most materials that could be used to demonstrate longitudinal vibrations with low frequencies would have high damping. A marshmallow might be a good example. Its low density would lead to a low frequency, but its internal sponginess would lead to the rapid damping of any waves in it.
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