sound levels

How is sound level quantified?At this point, you know that the loudness of a sound is

related to its amplitude.

A sound wave with a large amplitude sounds loud.

A sound wave with a small amplitude sounds quiet.

“Loud” and “quiet” are good relative terms, but in physics, we like to be more definite in our measurements.

How is sound level quantified?We’ve seen that the human ear is more sensitive to

sounds of specific frequencies. We are most sensitive to sounds at ~4000 Hz- we just hear these better.

Because of this, a 4000 Hz sound may appear louder than a 200 Hz sound, even if these waves actually have the same amplitude. This is just because our ears are better able to respond to the 4000 Hz wave.

How is sound level quantified?• Another problem with the “loud”/”quiet” system of

sound measurement is that individuals may have different standards for what sounds loud or quiet, or the loudness sensation could depend on the situation.

• For example, normal conversation might sound loud in a library, but would sound quiet at a concert.

• Humans are no better at being sound measurement devices than they are at being temperature measurement devices.

How is sound level quantified?To better express sound levels, two quantifiable

measurements were developed: Intensity, andDecibel level

These are different ways of measuring the sound loudness, and are related to each other. Knowing one, you can obtain the other.

How is sound level quantified?• In order to quantify sound level, there had to be a way

of measuring it.

• Sound loudness is related to amplitude, which is related to energy. Waves with large amplitudes are carrying a lot of energy. To be specific, the energy of a sound wave is proportional to the square of its amplitude- if one sound wave has twice the amplitude of another, it has four times the energy of the other.

How is sound level quantified?• The intensity of a sound is a measure of its energy,

but, more specifically, the rate at which energy is being delivered by the wave.

• Remember that energy per unit time, E/t, is a measure of power. Power is measured in watts, and 1 W = 1 J/s.

• So, sound intensity is a measure of sound power, but that gets more specific as well.

• Sound intensity measures how concentrated the sound wave’s power is.

How is sound level quantified?• A sound wave with a lot of power concentrated in a

small area will have a high intensity.

• Think of a powerful sound wave with a high concentration of energy hitting your eardrum. Your eardrum will experience a lot of force and vibrate back and forth widely. Your brain registers this as a loud sound. If we measure the power and its concentration, we’ll get a number with which we can quantify this loudness.

How is sound level quantified?• The concentration of a sound’s power is a

measure of how widely or narrowly it’s distributed over a given area.

• In equation form, we have:

where P is the wave’s power, in watts;A is the area over which the wave energy is being spread, in

m2; andI is the intensity, in W/m2€

I =P

A

How is sound level quantified?• We’d use this equation in a situation like

this:

A sound wave has a power of 60 W spread out over a 100 m2 area. What is this sound wave’s intensity?Known: P = 60 WA = 100 m2

€

I =P

A=

60W

100 m2= 0.6W /m2

Sound Intensity and Distance

• You know from experience that the same sound can appear loud or quiet depending upon how far away you are from the source.

• If you’ve been to a concert or dance, you’ve noticed that it can be painfully loud near a speaker, but the sound is at a more manageable level further away from the speaker.

• Why?


This is because the sound’s power becomes less and less concentrated as it spreads out.

Each wave produced by a constant sound source has the same amount of energy, and, thus, the same amount of power.

We’ve seen, however, that waves expand outward in circles that get bigger and bigger as they move away from the source.


Let’s establish a reference by looking at a wave in water:


If a sound wave were 2-dimensional, its energy would spread out into ever-expanding circles.

Since sound waves in air are 3-dimensional, their energy spreads out into ever-expanding spheres.


Each sphere represents one wave, and each wave contains the same amount of energy/power.

As the spheres get bigger, the power becomes more spread out. Bigger spheres have more surface area, so the power is less concentrated.


Since I=P/A, as A increases and P stays the same, I must decrease.

This is why sounds are less intense the further away you are from the source.

Your ear receives a smaller portion of the wave’s power, so it’s not as loud.


We can use the intensity equation, and the equation for the surface area of a sphere to develop an equation relating the sound intensity to the distance from the source.

The source is at the center of the concentric spheres of sound waves. When you stand some distance from the source, that distance is the radius of the sphere of sound you’re receiving.

Sound Intensity and DistanceIf you’re 2 m from a sound source, you’re receiving a

wave that’s spread out over the area of a sphere with a 2 m radius.

The equation for calculating the surface area of a sphere is:

This gives us an intensity equation in this form:€

A = 4πr2

€

I =P

A=P

4πr2

Sound Intensity and Distance• So, for a spherical sound wave emitted from a point

source, the relationship between the sound intensity and the distance, r, from the source is:

• Note that this, like other physics equations, is an idealization. In reality, obstacles, echoes, reverberations, and other actions affect the actual sound intensity.€

I =P

4πr2

Sound Intensity and Distance• Did you notice the specific relationship between

intensity and distance?

• Observe the inverse-square relationship between intensity and distance, as with gravitational force.

• This shows that doubling the distance between you and a sound source will cause the sound intensity to decrease to a quarter of its original value. If you get 3 times closer, the intensity will be 9 times greater, and so on.

€

I =P

4πr2

Sound Intensity and DistanceThis image should be helpful. It comes from

hyperphysics.phy-astr.gsu.edu.

Sound Intensity and DistanceLet’s look at some examples.

A source emits sound waves with a power of 2 W. How intense is the sound 2 m away from the source? 20 m away from the source?

Known: P = 2 W; r1 = 2 m; r2 = 20 m

First, find the intensity 2 m away:

Then, we could run the calculation again, with r = 20 m, or note that we have increased the distance by 10 times. The inverse square relationship tells us that this would cause the intensity to decrease by 100 times. Either way, our new intensity is 0.0004 W/m2

€

I =P

4πr2=

2W

4π (20m)2= 0.0004W /m2

€

I =P

4πr2=

2W

4π (2m)2= 0.04W /m2

Sound Intensity and DistanceWe can also use the equation to solve for other variables it contains.

A sound has an intensity of 0.15 W/m2 at a location 1.2 m from the source. What is this wave’s power?

Known: I = 0.15 W/m2; r = 1.2 m

Note that this is the power that each emitted wave has. Knowing that, we could use this to find the intensity at any radial distance.

€

I =P

4πr2so

P = 4πIr2 = 4π (0.15W /m2)(1.2m)2 = 2.7W

What do the intensity values mean?

In the last problem, it was stated that the sound intensity was 0.15 W/m2, and in the example before that, the intensity was calculated to be 0.04 W/m2.

We can compare these numbers and know that the 0.15 W/m2 sound is louder than the 0.04 W/m2 sound, but are these loud sounds? Quiet sounds? We need some reference values to know.


Here’s a table of different sounds, along with their level in decibels, pressure value, and intensity (I). This comes from www.sengpielaudio.com.


• Compare the sounds from our examples to the reference values in the table.

• Ours would be somewhere between the sound of a chainsaw from 1 m away, and the “threshold of discomfort”- a level at which listening to the sound becomes physically uncomfortable.

• The values 0.04 and 0.15 seem like small numbers, but they represent loud sounds.


Note the range of intensity values in the table.

Sound first becomes audible at an intensity value of 0.0000000000001 W/m2. This is 1.0x10-12 W/m2.

This is called the “threshold of hearing”. Sounds with less than this intensity don’t have enough power to vibrate your eardrum, so you can’t hear them.


• The highest intensity on the chart is 100 W/m2.

• An intensity of 10 W/m2 is referred to as the threshold of pain. At this level, the sound is so intense that it is physically painful to listen to.

• Think about this- a typical household light bulb emits 60 W of power. A square meter of area is what you’d enclose if you used 4 meter sticks to form the sides of a square. If the energy of this light bulb were spread out over your square meter of area, the intensity would be 60 W/m2. A sound this intense would cause physical pain.


This chart represents sound levels audible to humans.

We have an amazing range of sound intensities to which we are sensitive- we can hear them.

Because of this wide range, we don’t really notice a change in loudness until there is a fairly significant change in intensity.

Another way of quantifying sound level

• To make the numbers more manageable, the decibel (dB) scale was developed.

• Rather than spanning from 1x10-12 to 100, the decibel scale values go from 0 to 140. (There’s no reason sound levels can’t go higher than these high values, there’s just not much practical purpose for going higher)

• A sound level in W/m2 can be converted to its decibel equivalent.


• The decibel scale is a way of comparing how loud a sound is in comparison to the quietest sound you could hear.

• The decibel scale is logarithmic, so:Increasing the sound intensity by a factor ofo 10 raises its level by 10 dBo 100 raises its level by 20 dBo 1,000 raises its level by 30 dBo 10,000 raises its level by 40 dB o and so on(information copied from www.engineeringtoolbox.com)


• The equation for converting a sound intensity value to its equivalent dB value is this:

where I is the intensity value being converted, in W/m2;

Io is the intensity value at the threshold of hearing- this number is always 1 x 10-12 W/m2; and

L is the sound level in dB

€

L =10logI

Io

⎛

⎝ ⎜

⎞

⎠ ⎟


• Here’s a useful list for reference, thanks to www.engineeringtoolbox.com:

Sound intensity and feeling of loudness:

* 110 to 225 dB - Deafening * 90 to 100 dB - Very Loud * 70 to 80 dB - Loud * 45 to 60 dB - Moderate * 30 to 40 dB - Faint * 0 - 20 dB - Very Faint


• Let’s see how it’s used:

A sound has an intensity of 0.02 W/m2. What is this, in decibels? Is this a relatively loud or quiet sound?

Given: I = 0.02 W/m2

According to our chart on Slide 25, this would be louder than what you’d hear standing 1 m away from a speaker at a disco (!). According to the table on Slide 33, this is between “loud” and “deafening”. So, clearly, this would be sensed as a relatively loud sound.

€

L =10 logI

Io

⎛

⎝ ⎜

⎞

⎠ ⎟=10 log

I

Io

⎛

⎝ ⎜

⎞

⎠ ⎟=10 log

I

1x10−12W /m2

⎛

⎝ ⎜

⎞

⎠ ⎟

=10 log0.02W /m2

1x10−12W /m2

⎛

⎝ ⎜

⎞

⎠ ⎟=10 log(2x1010)

=10(10.3) =103dB

Another way of quantifying sound levels

A couple of notes on the dB equation: follow the example given and make sure you can

obtain the same answer- it may have been a while since you’ve used the log function, and dividing a number by a value with an exponent can be tricky

note that the number obtained doesn’t have units, technically- we tack on the dB unit to have a way of expressing the value, but the units don’t come from the input values

In Summary:

• Sound levels are quantifiable.

• As sound moves away from its source, it maintains the same energy and power, but the power becomes less concentrated as it is spread over spherical areas that get larger and larger.

• There is an inverse-square relationship between sound intensity and the distance away from the source.

• Sound intensity is a measure of a sound’s power per unit area.

• Sound intensity is quantified in units of W/m2. This value can be converted to decibels for an alternate measure of a sound’s level.

In Summary:

• Audible intensity values range from a low of 10-12 W/m2 to a high of 100 W/m2 (ranges are approximate).

• Audible decibel levels range from 0 dB to 140 dB (again, these are approximations)

• The low value, 10-12 W/m2 (0 dB), is known as the threshold of hearing- this is the level at which the typical person with good hearing can first sense that a sound exists.

• Certain high values, 10 W/m2 (130 dB), are known as the threshold of pain- this is the level at which sounds cause physical pain to the listener.

In Summary:

The equation for sound intensity is:

The equation relating sound intensity to the distance from the source is:

And, the conversion of a sound level from W/m2 to dB is done using this equation:

€

I =P

A

€

I =P

4πr2

€

L =10logI

Io

⎛

⎝ ⎜

⎞

⎠ ⎟

sound levels

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