# Chapter 11: Vibrations and Sound

**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 Summary

- Waves may interfere
*constructively*or*destructively,*giving rise to increases and decreases of the amplitude. - 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*}. - 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.
- The range of human hearing is 20 Hz-20,000Hz. Frequencies higher than this are called
*ultrasonic*frequencies. - 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.
- 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. - 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*}.
- 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*}.
- 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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