## Sound

3-15-99

Sections 12.1 - 12.4

### The speed of sound

A sound wave propagates by alternately compressing and expanding the medium. The speed of sound thus depends on how easily a medium can be compressed (or, equivalently, expanded). Sound travels fastest in media which are hard to compress (like different metals) because if one particle moves in response to a pressure wave, its neighbor will respond quickly. In easily-compressible media, such as most gases, the speed of sound is slower because particles respond more slowly to motion of neighboring particles.

The compressibility of a material is measured by the bulk modulus, discussed in chapter 9. The speed of sound also depends on density, and is given by:

For an ideal gas, it turns out that the speed of sound is given by:

m is the mass of one molecule.

In air at 0 °C, the speed of sound is 331 m/s. It increases with temperature. In liquids, the speed of sound is more like 1000 m/s, and in a typical solid more like 5000 m/s.

### Intensity

Just as the total energy in an oscillating spring is proportional to the square of the amplitude of the oscillation, the energy in a sound wave is proportional to the square of the amplitude of the pressure difference. A sound wave is usually characterized by the power (energy / second) it carries: the intensity is the power (P) divided by the area (A) the sound power passes through:

The human ear can detect sound of very low intensity. The smallest detectable sound intensity, known as the threshold of hearing, is about 1 x 10-12 W / m2. Sound that is 1 W / m2 is intense enough to damage the ear.

If sound moves away from a source uniformly in all directions, the intensity decreases the further away from the source you are. In fact, the intensity is inversely proportional to the square of the distance from the source. At a distance r away from a source sending out sound with a power P, the sound passes through a sphere with a surface area equal to . The intensity is thus:

This dependence on 1/r2 applies to anything emitted uniformly in all directions (sound, light, etc.).

### Decibels

The human ear has an incredibly large range, being able to detect sound intensities from 1 x 10-12 W / m2 to 1 W / m2. A more convenient way to measure the loudness of sound is in decibels (dB); in decibels, the range of human hearing goes from 0 dB to 120 dB. The ear responds to the loudness of sound logarithmically, so the decibel scale is a logarithmic scale:

On the decibel scale, doubling the intensity corresponds to an increase of 3 dB. This does not correspond to a perceived doubling of loudness, however. We perceive loudness to be doubled when the intensity increases by a factor of 10! This corresponds to a 10 dB increase. A change by 1 dB is about the smallest change a human being can detect.

### An example using decibels

A particular sound has an intensity of 1 x 10-6 W / m2 . What is this in decibels? If the intensity is increased by 15 dB, what is the new intensity in W / m2?

The intensity in dB can be found by simply applying the equation:

If this is increased to 75 dB, the new intensity can be found like this:

Therefore, I / Io = the inverse log of 7.5. Taking the inverse log of 7.5 means simply raising 10 to the 7.5 power, so:

This is 31.6 times as much as the original intensity.

### The ear

The decibel scale is used because it closely corresponds to how we perceive the loudness of sounds. The human ear is really quite an amazing detector of sound, and it's worth spending some time learning how it works.

The ear is split into three sections, the outer ear, the middle ear, and the inner ear. The outer ear acts much like a funnel, collecting the sound and transferring it inside the head down a passage that's about 3 cm long, ending at the ear drum.

The ear drum separates the outer ear from the middle ear, and, much like the skin on a drum, it is a thin membrane that vibrates in response to a sound wave. The middle ear is connected to the mouth via the eustachian tubes to ensure that the inside of the eardrum is maintained at atmospheric pressure. This is necessary for the drum to be able to respond to the small variations in pressure from atmospheric pressure that make up the sound wave.

In the middle ear are three small bones, called the hammer, anvil, and stirrup because of their shapes. These transfer the sound wave from the ear drum to the inner ear. Similar to a hydraulic lift, the pressure is transferred from a relatively large area (the eardrum) to a smaller area (the window to the inner ear). By Pascal's principle (see the section on fluids), the pressure is constant. The force is smaller at the small-area inner ear, but the work done at each end is equal, so the inner ear experiences a vibration with a much larger amplitude than that at the ear drum. The bones, in effect, act as an audio amplifier.

The eardrum and the window to the inner ear have different acoustic properties. If they were directly connected together some energy would be reflected back. The three bones in the middle ear are designed to transfer sound energy from the eardrum to the inner ear without any energy lost to reflections. The technical physics term for this is "impedance match": any time energy is transferred from one system to another without any reflected energy, the impedances are matched at the transfer point...in this case, the bones provide the impedance matching. The bottom line is that they will amplify the sound level without losing any sound energy.

The inner ear contains a fluid-filled tube, the cochlea. The cochlea is coiled like a snail, is about 3 mm in diameter, and is divided along its length by the basilar membrane. It also contains a set of hair cells that convert the sound wave into electrical pulses; these are transferred along nerves to the brain, to be interpreted as sound. When a sound signal enters the inner ear, a small movement of the basilar membrane or the fluid in the cochlea results in the rubbing of another membrane across the hair cells. The relatively long hairs provide another level of amplification, in the sense that a small force applied at the ends is converted into a relatively large torque.

To summarize, then, the outer ear collects sound and transfers it to the middle ear; the middle ear amplifies the sound and passes it to the inner ear; and the inner ear converts the sound into electrical signals to be sent to the brain.

The range of human hearing is quite large, both in terms of sound intensity and sound frequency. Humans can hear sounds between about 20 Hz and 20000 Hz; music and speech typically covers the range from 100 Hz to 3000 Hz. The ear is most sensitive to sound of about 3000 Hz, as the following graph shows: