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# Chapter 11: Vibrations and Sound

Difficulty Level: At Grade Created by: CK-12

Waves are periodic motion that induces motion beside it.  One model for this is a crowd wave, where you stand up and sit back down when the person beside you does.  A similar transfer happens when a rope is snapped or sound emitted.  This chapter covers the properties of waves as they move through a physical medium, focusing on sound in particular.

Chapter Outline

### Chapter Summary

1. Waves may interfere constructively or destructively, giving rise to increases and decreases of the amplitude.
2. Wave interference is responsible for the phenomenon of beats; where the amplitude of the sound changes when two sounds with close frequencies \begin{align*}f_1\end{align*} and \begin{align*}f_2\end{align*} are produced. The resulting beat frequency is \begin{align*}f_b = f_2 - f_1\end{align*}.
3. The intensity of a wave is defined as the average power produced by some source divided by the surface area over which the energy is transmitted. The equation for the intensity is \begin{align*}I = \frac{W}{A}\end{align*} where \begin{align*}I\end{align*} represents intensity, \begin{align*}W\end{align*} power (the wattage) and \begin{align*}A\end{align*}the area over which the energy, in a given amount of time, has spread.
4. The range of human hearing is 20 Hz-20,000Hz. Frequencies higher than this are called ultrasonicfrequencies.
5. The wave velocity equation is \begin{align*}v = \lambda f\end{align*} where \begin{align*}v\end{align*} is the velocity with which the wave travels; \begin{align*}\lambda\end{align*} is the wavelength of the wave; and \begin{align*}f\end{align*}is the frequency of the wave.
6. Many objects possess a natural or resonant frequency with which they vibrate. Strings and pipes (tubes) have multiple resonant frequencies as do most musical instruments. During resonance the amplitude of vibration of an object increases dramatically.
7. Both fixed strings and open pipes have resonant wavelengths of \begin{align*}\lambda_n = \frac{2}{n}L\end{align*}, where \begin{align*}L\end{align*} is the length of the string that is in resonance, and \begin{align*}n = 1,2,3\ldots\end{align*}.
8. Both strings fixed at only one end and closed pipes have resonant wavelengths of \begin{align*}\lambda_n = \frac{4}{n}L\end{align*} where \begin{align*}L\end{align*} is the resonant length of pipe or tube, and \begin{align*}n = 1,3,5 \ldots\end{align*}.
9. The Doppler effect occurs when a source of sound and/or the receiver are in motion.

\begin{align*}f' = \frac{v '}{\lambda '} = f \left( \frac{v + v_r}{v + v_s} \right)\end{align*}

A source moving at \begin{align*}v_s\end{align*} changes the wavelength.  The source moving away stretches out the wavelength, giving positive \begin{align*}v_s\end{align*}.

A receiver moving at \begin{align*}v_r\end{align*} changes the effective wave speed.  The receiver moving towards the source makes the waves arrive faster, giving positive \begin{align*}v_r\end{align*}.

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Date Created:
Jun 27, 2013