### Summary

## 17.1 Sound Waves

- Sound is a disturbance of matter (a pressure wave) that is transmitted from its source outward. Hearing is the perception of sound.
- Sound can be modeled in terms of pressure or in terms of displacement of molecules.
- The human ear is sensitive to frequencies between 20 Hz and 20 kHz.

## 17.2 Speed of Sound

- The speed of sound depends on the medium and the state of the medium.
- In a fluid, because the absence of shear forces, sound waves are longitudinal. A solid can support both longitudinal and transverse sound waves.
- In air, the speed of sound is related to air temperature
*T*by$v=\phantom{\rule{0.2em}{0ex}}331\frac{\text{m}}{\text{s}}\sqrt{\frac{{T}_{\text{K}}}{273\phantom{\rule{0.2em}{0ex}}\text{K}}}=331\frac{\text{m}}{\text{s}}\sqrt{1+\frac{{T}_{\text{C}}}{273\text{\xb0}\text{C}}}.$ *v*is the same for all frequencies and wavelengths of sound in air.

## 17.3 Sound Intensity

- Intensity $I=P\text{/}A$ is the same for a sound wave as was defined for all waves, where
*P*is the power crossing area*A*. The SI unit for*I*is watts per meter squared. The intensity of a sound wave is also related to the pressure amplitude $\Delta p\text{:}$$$I=\frac{{(\text{\Delta}p)}^{2}}{2\phantom{\rule{0.2em}{0ex}}\rho v},$$where $\rho $ is the density of the medium in which the sound wave travels and ${v}_{w}$ is the speed of sound in the medium. - Sound intensity level in units of decibels (dB) is
$$\beta (\text{dB})=10\phantom{\rule{0.2em}{0ex}}{\text{log}}_{10}\left(\frac{I}{{I}_{0}}\right),$$where ${I}_{0}={10}^{\mathrm{-12}}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$ is the threshold intensity of hearing.
- The perception of frequency is pitch. The perception of intensity is loudness and loudness has units of phons.

## 17.4 Normal Modes of a Standing Sound Wave

- Unwanted sound can be reduced using destructive interference.
- Sound has the same properties of interference and resonance as defined for all waves.
- In air columns, the lowest-frequency resonance is called the fundamental, whereas all higher resonant frequencies are called overtones. Collectively, they are called harmonics.

## 17.5 Sources of Musical Sound

- Some musical instruments can be modeled as pipes that have symmetrical boundary conditions: open at both ends or closed at both ends. Other musical instruments can be modeled as pipes that have anti-symmetrical boundary conditions: closed at one end and open at the other.
- Some instruments, such as the pipe organ, have several tubes with different lengths. Instruments such as the flute vary the length of the tube by closing the holes along the tube. The trombone varies the length of the tube using a sliding bar.
- String instruments produce sound using a vibrating string with nodes at each end. The air around the string oscillates at the frequency of the string. The relationship for the frequencies for the string is the same as for the symmetrical boundary conditions of the pipe, with the length of the pipe replaced by the length of the string and the velocity replaced by $v=\sqrt{\frac{{F}_{T}}{\mu}}.$

## 17.6 Beats

- When two sound waves that differ in frequency interfere, beats are created with a beat frequency that is equal to the absolute value of the difference in the frequencies.

## 17.7 The Doppler Effect

- The Doppler effect is an alteration in the observed frequency of a sound due to motion of either the source or the observer.
- The actual change in frequency is called the Doppler shift.

## 17.8 Shock Waves

- The Mach number is the velocity of a source divided by the speed of sound, $M=\frac{{v}_{\text{s}}}{v}.$
- When a sound source moves faster than the speed of sound, a shock wave is produced as the sound waves interfere.
- A sonic boom is the intense sound that occurs as the shock wave moves along the ground.
- The angle the shock wave produces can be found as $\text{sin}\phantom{\rule{0.2em}{0ex}}\theta =\frac{v}{{v}_{\text{s}}}=\frac{1}{M}.$
- A bow wake is produced when an object moves faster than the speed of a mechanical wave in the medium, such as a boat moving through the water.