Sound Intensity

Sound intensity, denoted I, measures the acoustic energy carried by a wave; the higher it is, the louder the sound. It is expressed in watts per square meter (W·m⁻²). It should not be confused with the sound intensity level L, in decibels (dB), which is what a sound level meter measures.

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How to measure it in class

The FizziQ app turns the smartphone’s microphone into a sound level meter and displays the sound level in decibels, in real time or as a graph.

Steps:

Open FizziQ’s sound level meter and let the measurement stabilize in a quiet room: you record the background noise, often 35 to 45 dB
Set up a fixed sound source (a speaker playing a continuous sound) and measure the level at 1 m
Repeat at 2 m, 4 m and 8 m, keeping the same orientation of the phone
Plot the level L as a function of the distance r, then as a function of log(r)
Check the loss of about 6 dB for each doubling of the distance

Scientific activities on this topic

Many activities and projects on sound intensity can be done with FizziQ. These activities can aim to better understand the concept of intensity, or to study the effects of loud sound on health:

Learn more

Why a logarithmic scale?

The human ear perceives intensities spanning twelve orders of magnitude: from the threshold of hearing (10⁻¹² W·m⁻²) to the threshold of pain (1 W·m⁻²), the ratio is one trillion. A linear scale would be unusable. We therefore define the sound intensity level:

L = 10 × log(I / I₀), with I₀ = 10⁻¹² W·m⁻²

This decibel scale compresses the twelve orders of magnitude into an interval from 0 to 120 dB, which is far more convenient. It also has the advantage of roughly matching perception: the ear responds approximately logarithmically to the stimulus.

Decibels do not add up

This is the most frequent error. Two identical sources each emitting 60 dB do not produce 120 dB, but about 63 dB. It is the intensities that add up, not the levels: doubling I amounts to adding 10 × log(2) ≈ 3 dB.

Be careful, though: this +3 dB rule assumes that the two sources are incoherent (two independent noises, two distinct instruments). If the two sources emit the same sound in phase, it is the amplitudes that add up, and the gain reaches 6 dB. This is also what makes anti-noise possible, where two waves in phase opposition cancel each other out.

Decay with distance

A point source radiating a power P in all directions spreads this power over a sphere of radius r, of area 4πr². Hence:

I = P / (4πr²)

The intensity therefore decreases as the inverse of the square of the distance. In decibels, doubling the distance means losing 10 × log(4) ≈ 6 dB. This law assumes a free field: in a classroom, reflections off the walls and ceiling limit the observed decay, and one often measures less than 6 dB.

Weighted decibels (dBA)

The ear is not equally sensitive to all frequencies: it is not very sensitive to low frequencies and is most sensitive around 3 to 4 kHz. Sound level meters therefore often apply an A-weighting, which corrects the spectrum to come closer to the sensation. The levels are then written dB(A). These are the ones used as a reference in occupational health.

Does adding two sounds really increase the sound level by 3 decibels? (FizziQ blog) details the case of coherent and incoherent sources.

Formula

Sound intensity level:

L = 10 × log(I / I₀)

Intensity at distance r from a point source (free field):

I = P / (4πr²)

Level difference between two points:

L₂ − L₁ = 10 × log(I₂ / I₁)

where:

I: sound intensity (W·m⁻²)
I₀: reference intensity, 10⁻¹² W·m⁻² (threshold of hearing at 1 kHz)
L: sound intensity level (dB)
P: acoustic power of the source (W)
r: distance to the source (m)

Some useful reference points:

+3 dB → intensity doubled
+10 dB → intensity multiplied by 10, sensation of a sound “twice as loud”
−6 dB → distance doubled in free field

Application examples

Threshold of hearing: 0 dB; rustling leaves: 20 dB; conversation: 60 dB; busy street: 80 dB; concert: 100 to 110 dB; threshold of pain: 120 dB
French regulations set 85 dB(A) over 8 hours as the threshold beyond which the employer must provide hearing protection: beyond that, the risk of irreversible hearing loss becomes significant
Noise barriers along highways exploit absorption and diffraction to gain 5 to 10 dB for nearby residents
An active noise-cancelling headset generates a wave in phase opposition with the ambient noise, effective mainly at low frequencies
Sonar and medical ultrasound imaging rely on measuring the intensity of a reflected wave

FAQ

Q: Why does FizziQ display decibels and not W·m⁻²? A: Because a microphone measures a pressure variation, from which a level relative to a reference is deduced. Converting to absolute W·m⁻² would require a precise calibration of the microphone, specific to each phone model. The decibel is in any case the quantity used in practice.

Q: Do two speakers each set to 70 dB give 140 dB? A: No, about 73 dB. It is the intensities that add up, not the levels. Doubling the intensity adds 3 dB.

Q: Why don’t I measure exactly −6 dB when I move to twice the distance? A: The 1/r² law assumes a free field, without reflection. In a room, the sound reflected by the walls adds to the direct sound and the measured decay is smaller. Outdoors, far from any wall, you do get close to the 6 dB.

Q: Is a 0 dB sound silence? A: No. 0 dB corresponds to I = I₀, the conventional threshold of hearing: it is a very faint sound but audible to a young ear at 1 kHz. Total silence would correspond to I = 0, a level tending toward minus infinity.

Q: What is the difference between sound intensity and sound power? A: The power P, in watts, characterizes the source itself and does not depend on where you stand. The intensity I, in W·m⁻², characterizes what arrives at a given point: it decreases as you move away.

Sound Level - Decibel - Inverse Square Law - Sound level meter - Active Noise Reduction