4.1 Logarithmic scale "dB"

In Chapter 1, we learned that humans can hear sound (audible sound) with a frequency range of around 20 Hz to 20 kHz and a sound pressure range of 20 µPa to 20 Pa and that the loudest sound is 10⁶ times louder than the least loud sound. Normally, logarithmic scales are used for efficiently presenting quantities with a wide range of dynamics. This also applies to sound pressure and noise as these can fluctuate across a wide range. The Weber-Fechner law states that the magnitude of a subjective sensation increases in proportion to the logarithm of the stimulus intensity. Auditory sense is a subjective sensation, which is why logarithmic scales are used to present sound.

Bel is used as the unit of logarithmic scales. This is because Alexander Graham Bell of the U.S. first used the unit to represent the attenuation of power transmission for telephone calls. As Bel is too lange for this purpose, in practice deci Bel (dB), which is one tenth of Bel, is used as the logarithmic scale unit. 1 B is equal to 10 dB.

The unit deci Bel (dB) is often used in the electrical and communications fields because, in addition to the reasons described above, it is easy to calculate the gain of multistage connected amplifiers, attenuators and other devices by addition and subtraction using the unit. As mentioned above, deci Bel (dB) is used as the unit for the attenuation (rate) of power transmission. It is defined as:

 

 

Where, E is measured power and E₀ is reference power.

As indicated by Equation 4-1, deci Bel (dB) is a relative value, defined in terms of the common logarithm of a power ratio. It can be handled as an absolute value if the reference value (e.g. E₀) is clearly defined. Whenever using deci Bel, it is important to clarify and keep in mind the reference value.

In the world of sound, the square of sound pressure is proportional to the intensity of sound. Based on this principle, sound pressure level L_P (dB), which is one of the units used by sound level meters, can be expressed by

 

 

A-weighted sound pressure level L_A (dB) can be expressed by the following equation using A-weighted sound pressure p_A. (For details, refer to Section 8.3, "Frequency correction circuit".)

 

 

In the world of sound (vibration), the term "sound (pressure) level" (dB) is used to express the loudness (intensity) of sound. In addition, as shown above, the absolute reference sound pressure for dB is clearly defined as p₀ (20 µPa). For example, the root mean square sound pressure for, say, 100 dB is determined to be 2 Pa. Calculation methods using deci Bel will be explained later in this guide.

 

5.1 Sound pressure level

Sound pressure level L_p (dB) can be expressed as:

 

where, p (Pa) is the root mean square value of instantaneous sound pressure of a given sound and p₀ (Pa) is the reference sound pressure.

 

The reference sound pressure p₀ in the atmospheric environment is 20 µPa, which is roughly the minimum audible value for a pure tone at 1 kHz for people with normal hearing. Figure 5-1 shows the relationship between sound pressure p (Pa) and sound pressure level L_p (dB). Sound pressure of 20 µPa equals sound pressure level of 0 dB. Sound pressure of 1 Pa equals sound pressure level of 94 dB. Sound pressure of 20 Pa equals sound pressure level of 120 dB. If there is a change of 0.1 atm (roughly 100 hPa = 10⁴ Pa), the sound pressure level will be 174 dB.

5-2 Sound intensity level

The energy I (W/m²) of sound passing through a unit area per unit time that is perpendicular to the direction of the sound waves can be expressed by:

 

where, the root mean square value of sound pressure being propagated through space is p (Pa) and the velocity of medium particles being vibrated by sound waves is u (m/s).

 

This quantity is called "sound intensity" I (W/m²).

 

With plane sound waves (whose surface is plane and perpendicular to the direction of propagation), the following equation is valid:

 

 

where, the medium’s volume density is ρ (kg/m³) and the sound velocity in the medium is c (m/s).

 

 

When the equation above is substituted into Equation 5-2, sound intensity I can be expressed as follows: Let us consider the following example. At 20 °C, the volume density of air ρ₀ is 1.205 kg/m³ and the velocity of sound c₀ is 343 m/s. Under these conditions, with plane waves that are being propagated through air, sound intensity I corresponding to the reference sound pressure p₀ = 2×10^-5 Pa can be expressed as follows:

 

 

Being almost equal to 10^-12 W/m², this value has been used, based on an international agreement, as the reference sound intensity I0 corresponding to the reference sound pressure p₀. As with sound pressure level, sound intensity level L_I (dB) can be defined as follows using the reference sound intensity I₀:

 

 

While the reference sound intensity I₀ was obtained for plane sound waves, Equation 5-6 for sound intensity level can also be used for ordinary sound waves. With plane sound waves where Equation 5-4 is valid, sound intensity level at 20 °C is calculated as follows:

 

 

This is almost equal to the sound pressure level. However, as the volume density of air is a function of temperature as is sound velocity, there is a difference of around 0.2 to 0.3 dB between sound pressure level and sound intensity level at temperatures outside 20 °C. At a point near a sound source and other certain points, sound waves are not regarded as plane waves, making it invalid to calculate sound intensity level simply based on sound pressure level.

5-3 Sound power level

Sound waves propagating in a medium can be regarded as a flow of energy. This energy is called sound energy. Sound energy passing through a unit area per unit time can be regarded as a quantity of sound energy and is called sound power P (W).

 

With plane sound waves, sound power P passing through a plane perpendicular to the direction of sound propagation can be calculated as follows:

 

where, p (Pa) is the root mean square sound pressure, ρ (kg/m³) is the volume density of the medium, c (m/s) is sound velocity and S (m²) is the area of the plane.

 

The quantity of sound power P expressed as a level relative to a reference value P₀ is called sound power level L_W (dB) and can be defined as follows:

 

 

The reference sound power level P₀ is 10^-12 W, which is the reference value I₀ (10^-12 W/m²) at sound intensity level L_I multiplied by a unit area. Sound power is primarily used to express the intensity of sound energy radiating from a sound source. The total sound energy radiating from a sound source per unit time within a specified frequency range is called sound output (sound power of a sound source) P (W), and the level of the sound power is called sound output level (sound power level of a sound source) L_W (dB).

5-4 Octave band level, 1/3 octave band level

To understand the physical properties of a sound, it is not enough to clarify its overall sound pressure and intensity levels. It is also necessary to clarify the sound pressure and intensity levels at each frequency (i.e. frequency analysis). Constant ratio filters such as octave band filters and 1/3 octave band filters are commonly used for frequency analysis of sound. These analyzers have been reduced in size and price thanks to active filtering for integrated circuits and other electronic circuit technologies, helping to improve measuring accuracy and ease of use. Octave band filters will be explained in detail in Chapter 11.