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# Physics Tutorial 2: The Physics of Hearing

by Mike Duda August 29, 2004

The most important mechanical wave to our everyday lives is sound. So how exactly do we hear? What is the definition of sound?

The Definition of Sound
The first thing we should understand about sound is that it is a mechanical wave. Any mechanical wave is a disturbance that travels through some material or substance, such as air, called a medium for the wave. Sound waves can travel through any kind of gas, liquid or solid medium.

Sound waves usually travel out in all directions from the source of sound, with an amplitude that depends on the direction and distance from the source. As the sound wave travels through the medium, the particles that make up the medium undergo displacements. Longitudinal wave displacements are parallel to the direction the wave travels. That just means the particles will move (be displaced) in the same direction the wave is moving.

Below, Figure 1.0 shows a cylindrical volume of air with a cross section S . If there is no displacement of the volume of gas, the length in the neutral (un-displaced) position is D x . Now say a sound wave (longitudinal wave) travels (propagates) in the direction of the cylinder of air and moves (displaces) it to the left. The right end of the cylinder will move a distance y 1 in the direction of the wave propagation . The left end of the cylinder will move a different distance y 2 in the same direction. The difference, y 2 - y 1 , is the displacement of the cylinder from its neutral position, and the Amplitude, A, is the maximum displacement of the cylinder.

Cylindrical Volume of Air being Displaced (see figure above)

Sound waves may also be described in terms of variation of pressure at various points in space (our living room could be considered this space). A sinusoidal sound wave in air (sinusoidal in oscillation but longitudinal in direction), the pressure fluctuates above and below atmospheric pressure in a sinusoidal variation with the same frequency as the motions of the air particles.

How the Human Ear Works
The human ear operates by sensing pressure variations above and below atmospheric pressure. A sound wave entering the ear canal exerts a fluctuating pressure on one side of the eardrum; the air on the other side of the eardrum is at atmospheric pressure. The pressure difference on the two sides of the eardrum sets it into motion. Microphones operate on a similar principle.

When your eardrum is set into motion by this pressure difference, it oscillates which then in turn causes oscillation of three tiny bones (ossicles) in the middle ear. This oscillation is finally transmitted to the fluid-filled inner ear; the motion of the fluid disturbs hair cells within the inner ear, which transmit nerve impulses to the brain with the information that a sound is present.

The human ear is sensitive to sound waves in the frequency range from about 20 to 20,000 Hz, which is called the audible range. You may have heard the term sound range, but sound waves can also be above audible range (ultrasonic) and below audible range (infrasonic). An audiophile is only interested in sound waves in the audible range.

Not everyone has the same sensitivity to sound waves, and our sensitivity declines with age. To get the most out of our stereo equipment, the audio amplifier and speakers should come close to producing a flat response across the audio frequency spectrum. These two items, amplifier and speakers, may perhaps be the most important components in our stereo setup.

Perception of Sound Waves
The perception of sound by the listener is directly related to the physical characteristics of the sound wave. For a given frequency, the greater the pressure amplitude of a sound wave, the greater the perceived loudness.

An important factor that contributes is that the ear is not equally sensitive to all frequencies in the audible range. A sound at one frequency may seem louder than one of equal pressure amplitude at a different frequency.

The curves shown in Figures 2.0 indicate how the perceived loudness is a function of both the frequency and the amplitude of the sound wave. The unit used is the called a "phon". At 1 kHz the phon and the dB Sound Pressure Level (SPL) are identical. As frequency varies, the phon follows the contour curve while the dB remains constant.

These curves tell us in some frequency regions the sound wave amplitudes must be increased to be perceived as equally loud as a sound of 1 kHz, while other wave frequencies must have their amplitude attenuated for the same perception.

The frequency range your hearing accentuates coincides with the frequency range in which important lingual sounds have their major spectral contents. Sounds like "p" and "t" have very important parts of their spectral energy within the "accentuated" range, making them more easy to discriminate between. To hear sounds in the accentuated range is vital for speech communication.

Figure 2.0, Loudness Curves (Fletcher-Munson Curves)

If the curves are turned upside down, as in Figure 2.1, the curves tell give us an idea of how the human hearing attenuates and accentuates parts of the audible frequency range.

Figure 2.1, Inverted Loudness Curves

## Physics Tutorial 2: The Physics of Hearing - page 2

Pitch and Timbre
The frequency of a sound wave is the primary factor in determining the pitch of a sound, the quality that lets us classify the sound as "high" or "low". The higher the frequency of the sound within the audible range, the higher the pitch that a listener will perceive.

Pressure amplitude does play a role in determining pitch. When a listener compares two sinusoidal waves with the same frequency but different pressure amplitudes, the one with the greater pressure amplitude is usually perceived as louder but also as slightly lower in pitch .

Musical sounds have wave functions that are more complicated than a simple sine function; musical sounds consist of a fundamental frequency and harmonics. Harmonics are frequencies higher than the fundamental frequency. Two tones produced by different instruments might have the same fundamental frequency (and thus the same pitch) but sound different because of the presence of different amounts of various harmonics. The difference between the two sounds is called timber (tone color) and is often described in subjective terms such as reedy, golden, mellow or whatever else you can contrive. The same principle applies to the human voice.

Figure 2.0, Harmonic Content of Two Wind Instruments

Another factor in determining tone quality is the behavior at the beginning (attack) and end (decay) of a tone. A piano tone begins with a thump and then dies away gradually.

Noise is a combination of all frequencies, not just frequencies that are integer multiples of a fundamental frequency. One example is the sound of the wind.

Sound Intensity
Sound waves, like all other traveling waves, transfer energy from one region of space to another. Intensity of a wave is the time average rate at which energy is transported by the wave, per unit area, across a surface perpendicular to the direction of propagation. In other words, intensity is the average power per unit area.

Since sound is a wave, characterized by amplitude and frequency, the mathematical formula for sound intensity includes the product of the sound wave frequency and amplitude. Lower frequencies produce smaller sound intensities. This is the reason a low-frequency woofer has to vibrate with a much larger amplitude than a high-frequency tweeter to produce the same intensity.

Lets put this into a perspective. The average total sound power emitted by a person speaking in a conversational tone is about 10 -5 Watts , while a shout corresponds to about 3 x 10 -2 Watts . If all the residents in New York City were to talk at the same time, the total sound power would be about 100 Watts, equivalent to the electric power required for a medium-sized light bulb.

If a source of sound can be considered as a point, the intensity at a distance r from the source is inversely proportional to r 2 . This "inverse-square" relationship also holds for various other energy-flow situations with a point source, such as light emitted by a point source.

The inverse-square relationship between sound intensity and distance from the source does not apply indoors because sound energy can also reach a listener by reflection from the walls and ceilings.

Because the ear is sensitive over such a broad range of intensities, a logarithmic intensity scale is usually used. The sound intensity level ß of a sound wave is defined by the equation

ß = (10 dB) log I/I o

For this equation, I o is a reference intensity, chosen to be 10^-12 W/m^2 , approximately the threshold of human hearing at 1 kHz.

If the intensity of a sound wave equals I o , its sound intensity level is 0 dB. On the opposite end of the scale, the threshold of pain is about 120 dB (1 W/m^2 ).

Since the human ear is not equally sensitive to all frequencies in the audible range, some sound-level meters weight the various frequencies unequally. One such scheme leads to the so-called dBA scale; this scale de-emphasizes the low and very high frequencies, where the ear is less sensitive than at midrange frequencies.

Summary

Tool Three: Sound is a Longitudinal Wave

1. Any mechanical wave is a disturbance that travels through some material or substance, such as air. This substance is called a medium for the wave.
2. Sound waves may also be described in terms of variation of pressure at various points in space.
3. The human ear operates by sensing pressure variations above and below atmospheric pressure.
4. The human ear is sensitive to sound waves in the frequency range from about 20 to 20,000 Hz, which is called the audible range.

Tool Four: Sound Perception

1. The physical characteristics of a sound wave are directly related to the perception of that sound by the listener.
2. An important factor that contributes to perception is that the ear is not equally sensitive to all frequencies in the audible range.
3. The higher the frequency of the sound within the audible range, the higher the pitch that a listener will perceive
4. Musical sounds consist of a fundamental frequency and harmonics.
5. Sound waves, like all other traveling waves, transfer energy from one region of space to another.
6. The sound intensity level ß of a sound wave is defined by the equation

ß = (10 dB) log I/I o