Technical Report: What is a Sound Level Meter? (Part 2)

As discussed in Chapter 1, the frequency range of sounds that humans can perceive (audible sounds) is approximately 20 Hz to 20 kHz, and the sound pressure range is 20 μPa to 20 Pa, with the ratio of the sound pressure between the quietest and loudest sounds reaching as high as ^10⁶. Logarithmic scales are generally used as a measure that efficiently represents a wide range of changes, and because the range of change is so wide, this logarithmic scale is also used to represent the magnitude of sound pressure and noise. Furthermore, there is the Weber-Fechner law, which states that "the amount of human sensation is proportional to the logarithm of the amount of stimulus," and since hearing is one of these sensations, a logarithmic scale is used.

The bel (B) is used as a unit of logarithmic scale because it was first used by Alexander Graham Bell of the United States to express the attenuation of power transmission in telephones. However, since the value of a bel (B) itself is too large, the decibel (deci Bel = dB), which is one-tenth of a bel, is actually used. That is, 1 B is equal to a value of 10 dB.

(Note)

In addition to the reasons mentioned above, the decibel (dB) is frequently used in the electrical and telecommunications fields because the gains of multi-stage amplifiers and attenuators can be easily calculated by addition and subtraction.

As mentioned earlier, the decibel (dB) is used to express the transmission attenuation (ratio) of power, and its definition is as follows:

Equation 4-1

As can be seen from equation 4-1 above, decibels (dB) are relative values that express power ratios using common logarithms, and if the reference value (for example, _E0 above) is clearly defined, it can be treated as an absolute value. Therefore, when dealing with decibels, it is important to always pay attention to "what the reference value is."

In the field of acoustics, the square of sound pressure is proportional to the intensity of the sound. Therefore, the sound pressure level Lp (dB), which is one of the values displayed when measuring with a sound level meter, is given by the following formula:

Equation 4-2

Furthermore, the formula for calculating the noise level _LA (A-weighted sound pressure level) (dB) from the sound pressure _pA weighted with A-weighting, which is a frequency weighting corresponding to human hearing (see Chapter 8, Section 3, "Frequency Correction Circuits" for details), is as follows:

Equation 4-3

In the world of sound (vibration), the term "level" (with dB as the practical unit) is used to express its magnitude (intensity). Instead of saying "loudness," the "level of sound (pressure)" is expressed as "how many dB." Furthermore, as defined above, the absolute standard sound pressure value for dB is clearly defined as p0 (20 μPa). Therefore, the effective sound pressure value of a "100 dB sound pressure level" is uniquely determined in Pascals. In this example, 100 dB corresponds to 2 Pascals. The calculation method for decibels will be explained later.

(Note) About dB SPL
Sound pressure level is measured in SPL, and in the past, it was sometimes written as "dB SPL" to explicitly express the unit of sound pressure level, dB (decibels). However, recent JIS standards and laws such as the Measurement Act now consider simply writing dB to be the correct unit notation.

Under the old Measurement Law, the unit of measurement "hon" was used, but with the enforcement of the new Measurement Law in 1993, it was changed to the SI unit "dB" in accordance with the international standard ISO standard, and the use of the non-SI unit "hon" is now completely prohibited.

Furthermore, while the simultaneous use of "phon" and "dB" was permitted during a grace period until September 30, 1997, please note that from October 1, 1997 onwards, the unit has been completely standardized to "dB". Note that "phon" is a unit that represents a noise level equivalent to "dB," and although the unit names are different, they represent the same quantity (for example, 90 phon and 90 dB represent the same noise level).

(Note)
While dBA or dB(A) have been frequently used as units for noise level (A-weighted sound pressure level) for ease of understanding, current domestic and international standards (JIS and IEC) and the Measurement Law stipulate that the only unit permitted for both sound pressure level and noise level is "dB". Therefore, please be careful when using formal notation. As a basic principle, it is practical to "use 'dB' as the unit for all levels, and specify the type of quantity, such as Lp (sound pressure level) or LA (noise level)."

When the effective value of the instantaneous sound pressure of a sound is p (Pa), and the reference sound pressure is _p0 (Pa), the “sound pressure level” Lp (dB) is:

Equation 5-1

It is given by ;. The reference sound pressure _p0 is 20 μPa for sound in air, and is the minimum audible value for a 1 kHz pure tone for a human with nearly normal hearing. Figure 5-1 below shows the relationship between sound pressure p (Pa) and sound pressure level _Lp (dB), where a sound pressure of 20 μPa is a sound pressure level of 0 dB, 1 Pa is 94 dB, and 20 Pa is 120 dB. Although it is not an audible sound, if there is a pressure fluctuation of 0.1 atmospheres (approximately 100 hPa = ^10⁴ Pa), the sound pressure level will be 174 dB.

If the effective value of sound pressure propagating through space is p (Pa), and the particle velocity of the medium particles vibrating due to the sound wave is u (m/s), then the sound energy I (W/ ^m²) passing through a unit area perpendicular to the direction of sound wave propagation per unit time is:

Equation 5-2

It is given by semicolon. This quantity is called "sound intensity" I (W/ ^m²).

If a sound wave is a plane wave (a wave whose wavefront is a plane perpendicular to the direction of wave propagation), then, with the volume density of the medium being ρ (kg/ ^m³) and the speed of sound in the medium being c (m/s):

Equation 5-3

The following holds true. Substituting the above equation into equation 5-2, the sound intensity I is:

Equation 5-4

For example, at a temperature of 20°C, the volume density of air _ρ0 is 1.205 kg/ ^m³ and the speed of sound _c0 is 343 m/s. Therefore, for sound waves (plane waves) propagating through air, the sound intensity I corresponding to the standard sound pressure value p0 = 2 × ^10⁻⁵ Pa is:

Equation 5-5

This value is 10.^-12 W/m ² It is very close to the reference value of sound pressure p ₀ Reference value for sound intensity corresponding to I ₀ For this value (10^-12 W/m ² It is internationally agreed that this should be used.

Similar to sound pressure levels, the "level of sound intensity" L _I (dB) is as described above I ₀ Taking this as the reference sound intensity;

Equation 5-6

It is defined by a semicolon.

The reference value for sound intensity _I₀ was determined for plane sound waves, but equation 5-6 for sound intensity levels is also used for general sound waves. Furthermore, for plane sound waves, equation 5-4 holds true, so the sound intensity level at a temperature of 20°C is:

Equation 5-7

This results in a value that is approximately equal to the sound pressure level. However, since the volume density of air and the speed of sound are functions of temperature, when the temperature is other than 20°C, the sound pressure level and the sound intensity level will not match, resulting in a difference of about 0.2 to 0.3 dB. Also, near the sound source, sound waves can no longer be considered plane waves, so the sound intensity level cannot be simply determined from the sound pressure.

Sound waves propagating through a medium can be considered as a flow of energy, and this energy is called acoustic energy. Therefore, as a quantity that represents the magnitude of this acoustic energy, we consider the acoustic energy passing through a specified surface per unit time, and call this "acoustic power" P (W).

If a sound wave is a plane wave, the acoustic power P passing through a plane perpendicular to the direction of sound propagation is given by: p (Pa) being the effective value of the sound pressure, ρ (kg/ ^m³) being the volume density of the medium, c (m/s) being the speed of sound, and S (^m²) being the area of the plane;

Equation 5-8

It is given by; Also, the quantity that expresses the sound power P as a level relative to a certain reference value P ₀ is called the “sound power level” L _W (dB);

Equation 5-9

It is defined by;. Reference value for sound power level P ₀ is 10^-12 W is the sound intensity level L. _I Reference value I ₀ (10^-12 W/m ² It is the value obtained by multiplying the unit area by the unit area. Acoustic power is mainly used to express the magnitude of acoustic energy radiated from a sound source. Within a specified frequency band, the total acoustic energy radiated by the sound source per unit time is called the "acoustic output (acoustic power of the sound source)" P (W), and the level of that acoustic power is called the "acoustic output level (acoustic power level of the sound source)" L _W It's called (dB).

When trying to understand the physical properties of sound, it is not enough to simply look at the overall sound pressure level or intensity level; it is necessary to determine the sound pressure level and intensity level for each frequency (frequency analysis).

When analyzing sound frequencies, analysis using constant frequency ratio filters such as octave band or 1/3 octave band filters is widely performed. These analyzers have been miniaturized and made more affordable through electronic circuit technology such as integrated circuit (IC) active filters, which has greatly contributed to improved measurement accuracy and ease of use. Details of octave band filters will be explained in Chapter 11.