Loudness and Sound Intensity

FIELDS OF STUDY: Acoustics; Classical Mechanics; Fluid Mechanics

ABSTRACT: The perceived loudness of a sound is directly related to its intensity. Intensity is an objective measure of sound power in the air in a given place. It is expressed in watts per square meter. Higher-intensity sounds seem louder, but the sensory ability of the ear and brain limits the perceived loudness of sounds. Loudness is influenced by the frequency of a sound. Higher-frequency sounds seem louder and higher pitched.

PRINCIPAL TERMS

• amplitude: a quantifying wave property measured from a point of rest to a point of maximum displacement; related to sound power and intensity of sound waves.
• decibel: a logarithmic unit that describes the power of a given sound in relation to the threshold of human hearing; abbreviated dB.
• frequency: how often a complete wave cycle occurs, which is directly proportional to the energy content of a wave and inversely proportional to wavelength; measured in hertz (Hz).
• just-noticeable difference (JND): the amount a parameter, such as sound intensity, must be changed so that the difference is noticeable at least half the time.
• logarithm: the power to which a fixed numerical base (the default is 10) must be raised to produce a given number.
• phon scale: a unit of measure for the perceived loudness of sounds; compares all sounds to a baseline sound of 1,000 hertz (1 kilohertz) frequency.
• sensitivity: the ability of an ear or a mechanical device to pick up and interpret sound; highly sensitive devices will have a smaller just-noticeable difference values.
• sone: a unit for quantifying perceived loudness; one sone equals forty phons.
• wavelength: the distance between crests of a wave; all electromagnetic radiation is transmitted as waves, with longer wavelengths corresponding to lower frequencies and less energy and vice versa.

Hearing a Sound

What someone perceives as sound is actually the product of a complex process wherein a sound wave traveling through the air (or another medium such as water) interacts with the delicate biological machinery of the ear and is then interpreted by the brain.

The perceived loudness of a sound largely depends on the intensity of the sound. This is directly related to the amplitude of the sound wave. Amplitude is one of the three major properties used to describe all waves, the other two being wavelength and frequency. These properties apply to mechanical waves, which pass through a medium, such as when sound waves pass through air or water, or to electromagnetic waves, such as when light or heat pass through empty space. The amplitude of a sound wave is determined by the degree to which the wave displaces the medium it is traveling through.

Sound forms compression (longitudinal) waves in which the wave’s motion is parallel to the direction the energy is being transferred. Amplitude in compression waves is measured as the maximum displacement of the particles of the medium from their normal resting state. Simply put, amplitude reflects how much energy a wave is carrying.

Intensity and Loudness

Sound intensity defines the sound power per unit area at a given point in a medium, typically in watts per square meter (W/m²). In practical acoustics, however, sound pressure level—which is directly related to the sound power—is very often used to describe intensity. For this discussion, assume "intensity" is determined in terms of sound pressure. In either case, intensity is an objective measure independent of an individual’s ability to hear.

Loudness, on the other hand, is a perceived quality of the interaction between the sound waves in the air and the human ear as interpreted by the brain. Different people have different sensitivities to sound and can perceive varying differences in intensity or frequency.

Because loudness is a perceived quality of sound, it is more difficult to quantify than an objective quality like intensity or frequency. Just-noticeable difference (JND) is an important element to consider when measuring a perceived quality. JND is the minimum amount a quality or parameter must be changed in order to notice the change at least half of the time.

Several units and measurement scales have been developed to describe loudness while taking into account human perception and JND values for intensity.

Decibels: Relative Sound Power

The decibel (dB) is the most common unit used to describe sound intensity, sound pressure level, and loudness. A decibel, however, has no defined quantity. Instead, it is used to describe the relationship between two values of power. For example, a rocket during takeoff produces about one quadrillion times as much sound power as a human breath. However, in describing the relationship using decibels, which follow a logarithmic scale, there is a 190 dB difference: a rocket at take-off produces about 200 dB of sound and the human breath produces about 10 dB.

The logarithm (log) of a given number is the power to which a fixed base (the default is 10) must be raised to produce the given number. For instance, the log of 100 is 2 because 10² = 100. To calculate sound intensity in decibels [I(dB)], using intensity (I) as measured in watts per meter squared, the following equation is used, where I₀ represents the threshold value of sound intensity (in watts per meter squared) for human hearing:

I(dB) = 10 log (I / I₀)

The value of I₀ at the standard frequency of 1 kilohertz (1,000 hertz) is 10⁻¹² watts per meter squared. Because they describe a ratio, decibels only make sense relative to this base value. Typically, the threshold of human hearing (near silence) is set as zero, with decibel values determined relative to this baseline. Decibels have the added advantage that they roughly correspond to the JND difference in loudness for human hearing.

Some typical sounds and their decibel values include:

• silence: 0 dB
• whisper: 20 dB
• public library: 40 dB
• dishwasher: 80 dB
• thunderclap: 120 dB

Exposure to very powerful sounds can cause a variety of problems. The threshold for annoying noise is usually somewhere between 70 to 80 dB, with discomfort increasing with power. Any sound at 80 dB or higher can cause hearing damage with long-term exposure. At 110 dB, average humans will begin to experience physical pain.

Phons and Sones

The phon and sone were developed in pursuit of a true unit of loudness. The phon was developed experimentally by playing two sounds to listeners. Both sounds always had the same intensity as measured in decibels. However, one sound would have a frequency of 1,000 hertz (Hz) and the other sound could be of any frequency. In this way, a scale was developed that compensated for the human ear perceiving sounds of different frequency as sounds that differ in intensity despite the sounds being of the same intensity. The phon relates all sounds to a baseline curve of sounds with equal perceived loudness but at a frequency of 1,000 Hz.

The sone was also developed using experiments in sound perception. In the sone experiments, listeners were asked to adjust the intensity of a sound (decibels) until they perceived it to double in loudness. The experiment found that a ten-decibel increase in sound pressure level roughly corresponds to a doubling of loudness, and this relationship forms the basis of the sone scale.

Both the phon and sone are relatively specialized units. Converting between phons, sones, and decibels is complex, but the basic relationships between the three are useful:

• Loudness in phons is equal to sound intensity in decibels for sounds with 1,000 Hz frequency.
• Forty phons equal 1 sone, and every 10 phons thereafter equals a doubling in sone value.
• Doubling the sone value equals a doubling in loudness, which equals an increase of 10 phons, which equals an increase in sound intensity of 10 dB.

The "Rule of Thumb" for Loudness

The relationship between the objective measure of sound via intensity and the subjective, perceived measure of sound via loudness can be roughly understood using one simple "rule of thumb." To double the loudness of a sound, increase its intensity tenfold. In other words, ten alarm clocks sound twice as loud as one alarm clock that is making the same sound. This is not a precise rule that applies across all situations, but for everyday calculations it is quite accurate.

Sample Problem

With the rule of thumb for loudness in mind, find the change in perceived loudness when an alarm clock’s output changes from 4 decibels in intensity to 16 decibels in intensity. Assume the frequency of the alarm does not change.

Answer

Begin by calculating the change in intensity—final intensity (I_f) relative to its starting intensity (I_s) —or I_f / I_s. Then, express the two decibel values using the equation for decibels given above:

16 dB = 10 log (I_f / I₀)

4 dB = 10 log (I_s / I₀)

Use these expressions to compare the starting and final intensities:

16 dB – 4 dB = 10 log (I_f / I₀) – 10 log (I_s / I₀)

12 dB = 10 [log (I_f / I₀) – log (I_s / I₀)]

According to the rules for subtraction using logarithms, this can be further rewritten as:

12 dB = 10 log [(I_f / I₀) ÷ (I_s / I₀)]

The two I0 values cancel each other out, and the decibel unit can be left behind since only the dimensionless ratio is of interest. To solve for I_f / I_s, isolate the log and then take the inverse log of each side:

12 = 10 log(I_f / I_s)

1.2 = log(I_f / I_s)

10^1.2 = I_f / I_s

Use the rule of thumb to convert this change in intensity into a change in loudness can be written mathematically so that change in loudness (L_f / L_s) is expressed in terms of change in intensity (I_f / I_s):

The alarm will sound about 2.3 times louder at its final intensity than it did at its starting intensity.

Sound Amplitude in Everyday Life

An understanding of sound intensity and perceived loudness enables audio technicians and engineers to design devices that produce sounds ranging from the enjoyable output of an amplified guitar to the annoying-but-effective buzz of an alarm clock. It also allows for the creation of devices that can help protect human ears in otherwise dangerous sound environments.

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