Formula For Intensity Of Sound

Learning Objectives

By the cease of this section, you will be able to:

Ascertain intensity, sound intensity, and audio pressure level.
Calculate audio intensity levels in decibels (dB).

Photograph of a road jammed with traffic of all types of vehicles.

Effigy one. Noise on crowded roadways like this one in Delhi makes it hard to hear others unless they shout. (credit: Lingaraj K J, Flickr)

In a quiet forest, you can sometimes hear a single leaf autumn to the ground. Subsequently settling into bed, you lot may hear your blood pulsing through your ears. But when a passing motorist has his stereo turned upwardly, you cannot even hear what the person next to you in your car is proverb. We are all very familiar with the loudness of sounds and aware that they are related to how energetically the source is vibrating. In cartoons depicting a screaming person (or an animal making a loud noise), the cartoonist frequently shows an open mouth with a vibrating uvula, the hanging tissue at the back of the mouth, to suggest a loud sound coming from the throat Figure 2. High noise exposure is hazardous to hearing, and it is common for musicians to take hearing losses that are sufficiently severe that they interfere with the musicians' abilities to perform. The relevant physical quantity is sound intensity, a concept that is valid for all sounds whether or not they are in the audible range.

Intensity is defined to be the power per unit surface area carried by a wave. Power is the rate at which energy is transferred by the wave. In equation form, intensity I is [latex]I=\frac{P}{A}\\[/latex], where P is the power through an area A. The SI unit for I is W/mⁱⁱ. The intensity of a sound wave is related to its amplitude squared by the following relationship:

[latex]\displaystyle{I}=\frac{\left(\Delta{p}\right)^2}{two\rho{v}_{\text{w}}}\\[/latex].

Here Δp is the pressure variation or pressure amplitude (half the deviation between the maximum and minimum pressure in the sound moving ridge) in units of pascals (Pa) or Northward/m². (Nosotros are using a lower case p for force per unit area to distinguish it from power, denoted by P higher up.) The free energy (as kinetic energy [latex]\frac{mv^ii}{2}\\[/latex]) of an oscillating element of air due to a traveling sound wave is proportional to its amplitude squared. In this equation, ρ is the density of the material in which the sound wave travels, in units of kg/yard³, and v _w is the speed of sound in the medium, in units of m/due south. The force per unit area variation is proportional to the amplitude of the oscillation, and so I varies every bit (Δp)² (Figure ii). This relationship is consistent with the fact that the sound wave is produced by some vibration; the greater its force per unit area aamplitude, the more the air is compressed in the sound information technology creates.

Figure 2. Graphs of the gauge pressures in two sound waves of different intensities. The more than intense audio is produced past a source that has larger-amplitude oscillations and has greater pressure maxima and minima. Because pressures are higher in the greater-intensity sound, it can exert larger forces on the objects it encounters.

Sound intensity levels are quoted in decibels (dB) much more ofttimes than sound intensities in watts per meter squared. Decibels are the unit of selection in the scientific literature too as in the pop media. The reasons for this choice of units are related to how we perceive sounds. How our ears perceive sound tin can be more accurately described by the logarithm of the intensity rather than directly to the intensity. The sound intensity level β in decibels of a sound having an intensity I in watts per meter squared is divers to exist [latex]\beta\left(\text{dB}\right)=x\log_{ten}\left(\frac{I}{I_0}\right)\\[/latex], where I ₀ = 10⁻¹² W/grand² is a reference intensity. In detail, I ₀ is the lowest or threshold intensity of audio a person with normal hearing can perceive at a frequency of 1000 Hz. Sound intensity level is not the same as intensity. Because β is defined in terms of a ratio, information technology is a unitless quantity telling you the level of the sound relative to a fixed standard (10⁻¹² W/m², in this case). The units of decibels (dB) are used to point this ratio is multiplied past ten in its definition. The bel, upon which the decibel is based, is named for Alexander Graham Bell, the inventor of the telephone.

Table 1. Audio Intensity Levels and Intensities
Sound intensity level β (dB)	Intensity I(West/m^two)	Example/event
0	1 × 10^–12	Threshold of hearing at 1000 Hz
x	one × 10^–xi	Rustle of leaves
twenty	ane × 10^–x	Whisper at 1 yard distance
30	1 × ten^–9	Quiet home
40	one × ten^–viii	Average abode
50	1 × x^–7	Average office, soft music
60	one × 10^–vi	Normal conversation
70	1 × 10^–5	Noisy role, decorated traffic
eighty	1 × 10^–four	Loud radio, classroom lecture
90	1 × x^–3	Within a heavy truck; damage from prolonged exposure^[one]
100	i × 10^–2	Noisy manufacturing plant, siren at 30 m; damage from 8 h per twenty-four hour period exposure
110	1 × x^–i	Impairment from 30 min per twenty-four hour period exposure
120	1	Loud stone concert, pneumatic chipper at 2 m; threshold of hurting
140	1 × x²	Jet aeroplane at 30 m; astringent pain, damage in seconds
160	one × 10⁴	Bursting of eardrums

The decibel level of a sound having the threshold intensity of x⁻¹² W/1000² is β = 0 dB, considering log_xi = 0. That is, the threshold of hearing is 0 decibels. Table 1 gives levels in decibels and intensities in watts per meter squared for some familiar sounds.

Ane of the more hit things nearly the intensities in Table 1 is that the intensity in watts per meter squared is quite small for about sounds. The ear is sensitive to as little equally a trillionth of a watt per meter squared—even more than impressive when you realize that the surface area of the eardrum is only about one cmⁱⁱ, so that simply 10^–16 W falls on information technology at the threshold of hearing! Air molecules in a sound wave of this intensity vibrate over a altitude of less than i molecular diameter, and the gauge pressures involved are less than 10^–ix atm.

Another impressive feature of the sounds in Table 1 is their numerical range. Sound intensity varies past a factor of 10¹² from threshold to a sound that causes damage in seconds. Y'all are unaware of this tremendous range in audio intensity because how your ears answer can exist described approximately as the logarithm of intensity. Thus, sound intensity levels in decibels fit your feel better than intensities in watts per meter squared. The decibel scale is also easier to chronicle to because most people are more accustomed to dealing with numbers such every bit 0, 53, or 120 than numbers such every bit i.00 × 10^–eleven.

1 more than observation readily verified past examining Table 1 or using [latex]I=\frac{\left(\Delta{p}\right)^2}{two\rho{v}_{\text{due west}}}\\[/latex] is that each factor of 10 in intensity corresponds to x dB. For example, a 90 dB audio compared with a 60 dB sound is 30 dB greater, or 3 factors of 10 (that is, 10³ times) as intense. Another example is that if one sound is 10⁷ as intense as another, it is 70 dB higher. See Tabular array two.

Table 2. Ratios of Intensities and Corresponding Differences in Sound Intensity Levels
[latex]\frac{I_2}{I_1}\\[/latex]	β _two –β ₁
2.0	3.0 dB
v.0	7.0 dB
10.0	10.0 dB

Example i. Calculating Sound Intensity Levels: Audio Waves

Calculate the sound intensity level in decibels for a sound wave traveling in air at 0ºC and having a pressure level amplitude of 0.656 Pa.

Strategy

We are given Δp, so we tin can calculate I using the equation [latex]I=\frac{\left(\Delta{p}\right)^2}{\left(2\rho{5}_{\text{westward}}\right)^ii}\\[/latex]. Using I, nosotros tin calculate β straight from its definition in [latex]\beta\left(\text{dB}\right)=10\log_{10}\left(\frac{I}{I_0}\right)\\[/latex].

Solution

1. Place knowns: Sound travels at 331 grand/due south in air at 0ºC. Air has a density of i.29 kg/m³ at atmospheric pressure and 0ºC.

2. Enter these values and the pressure amplitude into [latex]I=\frac{\left(\Delta{p}\correct)^2}{ii\rho{v}_{\text{westward}}}\\[/latex]:

[latex]I=\frac{\left(\Delta{p}\right)^two}{2\rho{v}_{\text{w}}}=\frac{\left(0.656\text{ Pa}\right)^ii}{two\left(1.29\text{ kg/thousand}^3\right)\left(331\text{ chiliad/due south}\right)}=5.04\times10^{-4}\text{ W/k}^2\\[/latex]

3. Enter the value for I and the known value for I ₀ into [latex]\beta\left(\text{dB}\right)=10\log_{10}\left(\frac{I}{I_0}\correct)\\[/latex]. Calculate to find the sound intensity level in decibels:

x log_ten(five.04 × 10^eight) = 10(viii.70)dB = 87 dB.

Discussion

This 87 dB sound has an intensity five times as bang-up as an 80 dB sound. So a factor of five in intensity corresponds to a departure of 7 dB in sound intensity level. This value is true for whatever intensities differing by a cistron of five.

Example 2. Alter Intensity Levels of a Sound: What Happens to the Decibel Level?

Show that if i audio is twice equally intense equally another, it has a sound level near 3 dB higher.

Strategy

You are given that the ratio of two intensities is 2 to ane, and are then asked to find the divergence in their sound levels in decibels. You can solve this problem using of the properties of logarithms.

Solution

one. Identify knowns.

The ratio of the two intensities is 2 to 1, or:

[latex]\frac{I_2}{I_1}=2.00\\[/latex].

Nosotros wish to show that the difference in sound levels is about 3 dB. That is, we want to evidence

β ₂ −β _i = 3 dB.

Notation that

[latex]\log_{10}b-\log_{10}a=\log_{x}\left(\frac{b}{a}\right)\\[/latex].

ii. Utilise the definition of β to go:

[latex]\beta_{2}-\beta_{1}=ten\log_{10}\left(\frac{I_2}{I_1}\right)=10\log_{10}2.00=10\left(0.301\right)\text{ dB}\\[/latex]

Thus,

β _two −β _ane = 3.01 dB.

Word

This means that the ii sound intensity levels differ by 3.01 dB, or about 3 dB, as advertised. Note that considering only the ratio [latex]\frac{I_2}{I_1}\\[/latex] is given (and not the actual intensities), this result is true for any intensities that differ past a factor of 2. For case, a 56.0 dB sound is twice as intense equally a 53.0 dB audio, a 97.0 dB sound is one-half as intense as a 100 dB audio, then on.

It should be noted at this point that in that location is another decibel scale in apply, called the sound pressure level, based on the ratio of the pressure amplitude to a reference pressure. This scale is used particularly in applications where sound travels in h2o. It is beyond the scope of near introductory texts to care for this scale because information technology is not commonly used for sounds in air, just it is of import to note that very unlike decibel levels may be encountered when sound pressure levels are quoted. For instance, ocean noise pollution produced past ships may be as great every bit 200 dB expressed in the sound pressure level, where the more familiar sound intensity level we use hither would be something nether 140 dB for the aforementioned audio.

Take-Abode Investigation: Feeling Audio

Find a CD player and a CD that has stone music. Identify the player on a light table, insert the CD into the player, and start playing the CD. Place your hand gently on the table adjacent to the speakers. Increment the book and note the level when the tabular array only begins to vibrate as the rock music plays. Increment the reading on the volume control until information technology doubles. What has happened to the vibrations?

Cheque Your Agreement

Role 1

Draw how amplitude is related to the loudness of a audio.

Solution

Aamplitude is straight proportional to the experience of loudness. Equally aamplitude increases, loudness increases.

Part 2

Identify common sounds at the levels of 10 dB, 50 dB, and 100 dB.

Solution

10 dB: Running fingers through your hair.

fifty dB: Inside a repose home with no television or radio.

100 dB: Take-off of a jet plane.

Section Summary

Intensity is the same for a audio wave as was defined for all waves; information technology is [latex]I=\frac{P}{A}\\[/latex], where P is the ability crossing area A. The SI unit for I is watts per meter squared. The intensity of a sound wave is also related to the pressure level amplitude Δp, [latex]I=\frac{{\left(\Delta p\right)}^{2}}{2{\rho{5}}_{west}}\\[/latex], where ρ is the density of the medium in which the sound wave travels and v _{due west} is the speed of sound in the medium.
Sound intensity level in units of decibels (dB) is [latex]\beta \left(\text{dB}\correct)=\text{10}\log_{x}\left(\frac{I}{{I}_{0}}\right)\\[/latex], where I₀ = ten^–12 W/mⁱⁱ is the threshold intensity of hearing.

Conceptual Questions

Six members of a synchronized swim team wear earplugs to protect themselves confronting water pressure at depths, merely they can even so hear the music and perform the combinations in the water perfectly. One day, they were asked to leave the pool then the dive squad could do a few dives, and they tried to practice on a mat, merely seemed to take a lot more difficulty. Why might this be?
A community is concerned about a plan to bring train service to their downtown from the town'southward outskirts. The current sound intensity level, even though the rail 1000 is blocks away, is lxx dB downtown. The mayor assures the public that there will be a difference of only 30 dB in audio in the downtown area. Should the townspeople exist concerned? Why?

Problems & Exercises

What is the intensity in watts per meter squared of 85.0-dB audio?
The warning tag on a lawn mower states that it produces dissonance at a level of 91.0 dB. What is this in watts per meter squared?
A sound wave traveling in 20ºC air has a pressure level amplitude of 0.5 Pa. What is the intensity of the wave?
What intensity level does the sound in the preceding problem represent to?
What audio intensity level in dB is produced by earphones that create an intensity of 4.00 × x⁻² W/m²?
Show that an intensity of 10⁻¹² Due west/mⁱⁱ is the aforementioned as x^−xvi W/grand^two.
(a) What is the decibel level of a sound that is twice as intense as a 90.0-dB sound? (b) What is the decibel level of a audio that is one-fifth as intense as a 90.0-dB sound?
(a) What is the intensity of a sound that has a level 7.00 dB lower than a 4.00 × 10^−nine W/m² sound? (b) What is the intensity of a sound that is 3.00 dB higher than a 4.00 × 10⁻⁹ W/m² audio?
(a) How much more intense is a sound that has a level 17.0 dB college than some other? (b) If one sound has a level 23.0 dB less than another, what is the ratio of their intensities?
People with adept hearing can perceive sounds every bit low in level equally −viii.00 dB at a frequency of 3000 Hz. What is the intensity of this sound in watts per meter squared?
If a big housefly three.0 one thousand abroad from y'all makes a noise of forty.0 dB, what is the racket level of m flies at that distance, assuming interference has a negligible effect?
X cars in a circle at a boom box competition produce a 120-dB sound intensity level at the center of the circle. What is the boilerplate sound intensity level produced at that place by each stereo, assuming interference effects can be neglected?
The amplitude of a sound wave is measured in terms of its maximum gauge pressure. Past what factor does the amplitude of a sound moving ridge increase if the sound intensity level goes upwardly by 40.0 dB?
If a sound intensity level of 0 dB at 1000 Hz corresponds to a maximum gauge force per unit area (audio aamplitude) of x^−ix atm, what is the maximum approximate pressure in a 60-dB sound? What is the maximum gauge force per unit area in a 120-dB sound?
An eight-hour exposure to a sound intensity level of ninety.0 dB may crusade hearing harm. What energy in joules falls on a 0.800-cm-diameter eardrum then exposed?
(a) Ear trumpets were never very common, but they did aid people with hearing losses by gathering sound over a large surface area and concentrating it on the smaller area of the eardrum. What decibel increase does an ear trumpet produce if its sound gathering area is 900 cmⁱⁱ and the expanse of the eardrum is 0.500 cm², only the trumpet only has an efficiency of 5.00% in transmitting the sound to the eardrum? (b) Comment on the usefulness of the decibel increase plant in part (a).
Sound is more effectively transmitted into a stethoscope by direct contact than through the air, and information technology is further intensified by being concentrated on the smaller area of the eardrum. It is reasonable to assume that sound is transmitted into a stethoscope 100 times every bit effectively compared with transmission though the air. What, then, is the gain in decibels produced by a stethoscope that has a sound gathering area of xv.0 cm², and concentrates the sound onto two eardrums with a total expanse of 0.900 cmⁱⁱ with an efficiency of 40.0%?
Loudspeakers tin can produce intense sounds with surprisingly small-scale energy input in spite of their depression efficiencies. Calculate the power input needed to produce a 90.0-dB sound intensity level for a 12.0-cm-bore speaker that has an efficiency of one.00%. (This value is the sound intensity level right at the speaker.)

Glossary

intensity: the power per unit expanse carried by a wave

sound intensity level: a unitless quantity telling you the level of the sound relative to a fixed standard

acoustic level: the ratio of the pressure amplitude to a reference pressure

Selected Solutions to Problems & Exercises

1. 3.sixteen × 10⁻⁴ West/mⁱⁱ

3. 3.04 × ten⁻⁴ W/thouⁱⁱ

five. 106 dB

vii. (a) 93 dB; (b) 83 dB

9. (a) 50.ane; (b) v.01 × 10⁻³ or [latex]\frac{1}{200}\\[/latex]

eleven. seventy.0 dB

13. 100

xv. 1.45 × 10⁻³ J

17. 28.2 dB