sound intensity

You’ve probably heard about decibels (dB). This measure of sound intensity is used to compare the intensity of different sounds (of the same frequency). The decibel is a unit derived from the bel (B), which is a logarithmic scale that compares the intensity of a sound with the lowest intensity of sound that can be observed by a human being. The bel scale is often expressed in tenths of its unit, called decibels.

Before we continue, we suggest reading our article on what is sound , so that you can understand the content of this article more clearly.

How to calculate sound intensity?

The intensity of the sound can be calculated if we make the ratio of the power of a sound wave, that is, the amount of energy it emits every second, with the area whose normal line (a line that makes 90º with the surface) is defined by the direction of propagation of sound.

Since sound waves are propagated in a three- dimensional way , sound sources emit sounds in a spherical shape , so the area through which sound waves transfer their energy is proportional to r² (r – radius of the sphere) – the square of the distance between the observer and the emitting source . Consequently, we say that the sound intensity is inversely proportional to the square of the distance between the observer and the emitting source.

The following figure [1] helps us to understand why the sound intensity decreases with the square of the distance from the emitting source (in blue). Notice that at distance r there is only one square; at distances 2r and 3r, there are 4 and 9 squares, respectively.

→ Sound intensity formula

The formula used to calculate the intensity (I) of sound waves relates the power (P) of the emitting source to the circular area (S) formed between the sound source and the observer.

I – sound intensity (W/m²)

P – power of the emitting source (W)

S – area (m²)

r – radius of the spherical area (m)

Before we go any further, it is important that you understand the properties of the logarithmic scale, so go to our text on logarithm ( click here ) and review the subject. To write the sound intensity on the bel scale, we must use the following relationship:

I ₀ – threshold of audibility

I – observed sound intensity

Example:

Consider a sound wave of intensity equal to 9.8 x 10 ^-5 W/m². Check the calculation of the sound intensity of this wave in decibels:

To perform the above calculation, we use two properties of the logarithm: one in which the exponent “comes out” of the logarithm and one in which the multiplication of logarithms is equal to their sum:

According to our calculation, a sound wave of intensity 9.8 x 10 ^-5 W/m² is equivalent to a sound of approximately 80 dB. This sound intensity corresponds to the noise we hear when we are close to avenues with busy traffic.

But, after all, what does it mean to say that a sound has an intensity of 80 dB? It means that its intensity is 10 ⁸ times greater than the intensity of the weakest sound we can hear (audible threshold: 10 ^-12 W/m²), that is, 100 million times more intense.

Similarly, a 30 dB sound is 100 times louder than a 10 dB sound, since the difference between these sound intensities is 20 dB (10.10).

Sound intensity and amplitude

The intensity of sound waves is proportional to the amplitude of the sound wave, not its frequency , so we say that loud sounds are loud sounds, while low-intensity sounds are called weak sounds . High intensity sounds are capable of transferring large amounts of energy every second, which can cause hearing damage, for example.

Be careful not to confuse the sound qualities . Loud and low sounds refer, respectively, to high-frequency sounds (treble sounds) and low-frequency sounds (bass sounds). The weak and strong sounds are related to the intensity of the sound, that is, “high volume” and “low volume”.

bells and decibels

As sound intensity measures characterize very small numbers, we use a measure that relates them to the smallest sound that can be heard by humans. This measure is known as bel. This logarithmic measure was named after the American inventor Alexander Graham Bell .

In everyday applications, it is quite common for us to use one-tenth of a bel, the decibel. The calculation of sound intensity using the bel scale helps us to understand what sounds are like at different sound intensities, as well as their effects on the human body.

The bels and decibels scale is widely used to compare measures of intensity and energy, therefore, other quantities can be expressed according to this unit. An example of this use is the Richter scale , used to compare different intensities of earthquakes .

Sound intensity tables

Between the intensities of 0 dB and 50 dB, the sound is in the range of acoustic comfort . Higher intensity sounds, between 50 dB and 60 dB, cause discomfort and irritability. Sounds of intensity beyond 60 dB can cause health damage , such as hearing loss, anxiety, irritation, insomnia, etc. Finally, sounds approaching 120 dB cause physical pain in the ears and should be avoided whenever possible .

Check the table below, which relates the sound intensity with the possible negative effects on human health. The data displayed is in accordance with World Health Organization (WHO) guidelines:

There are national guidelines that are used to determine the number of hours and the maximum intensity of noise to which people, especially workers, can be exposed without harm to their health. Check the time limits for exposure to sound, according to its intensity:

The table above [2] indicates that there are time limits for being exposed to each sound intensity. Failure to observe these limits can cause damage to health.

Interesting facts about sound intensity

• The title of the most intense sound in all of history is attributed to the eruption of the Krakatoa volcano, which occurred on August 27, 1883, in Indonesia. Reports from the time indicate that the sound could be heard over 3500 km away, in Australia. Even at this distance, the sound heard was compatible with the firing of a rifle. Measurements made at the time indicated that, at a distance of 150 km from the volcano, a sound of 172 dB was heard, the equivalent of the explosion of 200 million tons of dynamite.
• Although they are not very powerful, the small area of ​​the ear canal makes the sound intensity emitted by the headphones easily exceed the range of 100 dB, so this type of device should not be used for prolonged intervals or even at maximum volume. , in order to avoid permanent hearing damage.

Exercises solved on sound intensity

Question 1) A sound source emits sounds of 10 ^-6 W/m². The intensity of this sound, on the decibel scale, is equal to:

a) 60 dB

b) 80 dB

c) 120 dB

d) 100 dB

e) 50 dB

Data : I ₀ = 10 ^-12 W/m²

Template : Letter A

Resolution:

To solve this exercise, we must use the decibel formula, which uses a logarithmic relationship:

According to the calculation performed, the intensity of this sound corresponds to an intensity of 60 decibels, so the correct alternative is the letter A.

Question 2) Using a decibel meter, a technician decides to measure the sound intensity inside and outside a SPA room, aiming at the comfort of the guests. Its measurements indicate sound intensities of 20 dB and 60 dB. With these sound intensity measurements, the technician notes in his table that the 60 dB sound, measured outside the spa room, is ____ times more intense than the 20 dB sound. Choose the alternative that correctly completes the gap left in the sentence:

a) 40

b) 3

c) 1,000

d) 10,000

e) 10

Template : Letter D

Resolution:

The difference of 1 bel (10 decibels) between two measurements indicates that the magnitude of one of them is an order of magnitude higher than the other, that is, it is 10 times greater. In this case, we are dealing with decibels, and the difference in decibels between the two measures of sound intensity mentioned in the statement is 40 dB, that is, 4 bels, therefore, four orders of magnitude (10 ⁴ ). Therefore, 60 dB sound is 10,000 times louder than 20 dB sound.

Question 3) As we move away from an emitting source, the intensity of sound waves decreases. At a distance d, a sound intensity I is measured. Disregarding any obstacle, what must be the measured sound intensity, as a function of I, for a distance of 4d?

a) I/4

b) I/16

c) 4I

d) 12I

e) 25I

Template : Letter B

Resolution :

As we know, the sound intensity is inversely proportional to the square of the distance between the emitting source and the observer. Thus, when we are at a distance 4 times greater than d, the sound intensity measured at that position must be 16 times less than that measured at distance d, so the correct alternative is the letter B.