Why does music sound quieter as you move away from a speaker, why does a 10 dB increase matter so much, and why do two nearby notes sometimes produce a pulsing wah-wah effect? This lesson links sound intensity, logarithmic decibels, and beats.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
If one workplace is measured at 90 dB and another at 100 dB, is the second place only “a little bit” more intense? Predict before we unpack what decibels actually mean.
Type your prediction below. You will revisit it at the end.
Write your prediction in your book. You will revisit it at the end.
Wrong: Momentum is not conserved in collisions with friction.
Right: Momentum is always conserved in isolated systems; friction is an external force, so the system must include the surface.
📚 Core Content
As sound spreads out from a source, the same acoustic energy must cover an ever-larger surface area, so the energy arriving at each unit of area falls with distance.
For a point source in free space, the wavefronts form concentric spheres. The surface area of a sphere is $4\pi r^2$, so the intensity $I$ — power per unit area — obeys $I = P/(4\pi r^2)$. This is the physical origin of the inverse square law. If you double your distance from the source, the sphere’s area quadruples and intensity becomes one quarter. If you triple the distance, the area grows nine-fold and intensity drops to one ninth.
This rapid drop explains why a whisper across a large classroom is inaudible, why stage speakers are angled toward the audience, and why sound barriers are judged by how much they increase the effective distance between a noise source and a listener.
| Distance change | Intensity change | Meaning |
|---|---|---|
| Double distance | 1/4 intensity | Much quieter than many people expect |
| Triple distance | 1/9 intensity | Large drop in received intensity |
| Half distance | 4× intensity | Much more intense near the source |
Decibels are logarithmic, so equal steps on the dB scale do not represent equal linear changes in intensity. Instead, each 10 dB interval marks a tenfold multiplication of intensity.
The decibel scale is base-10 logarithmic. A 10 dB increase means the sound intensity is exactly 10 times greater. A 20 dB increase means 100 times the intensity, and a 30 dB increase means 1000 times the intensity. This compresses an enormous range of physical intensities into manageable numbers. The threshold of hearing is defined as 0 dB, which is not the absence of sound but the faintest pressure variation a typical human ear can detect — about $10^{-12}\ \text{W/m}^2$.
Because the human ear itself responds roughly logarithmically to intensity, the decibel scale also matches our subjective sensation of loudness better than a linear scale would. That is why a modest-looking decibel jump can correspond to a dramatic rise in the actual energy reaching the ear.
Beats are periodic changes in loudness caused by the superposition of two sound waves with slightly different frequencies.
When two waves travel through the same medium at the same time, they interfere according to the principle of superposition. At moments when the crests align, the amplitudes add constructively and the sound grows louder. A short time later, the crest of one wave meets the trough of the other, producing partial cancellation and a quieter sound. This loud-soft-loud pattern repeats at the beat frequency, which is the absolute difference between the two source frequencies.
Mathematically, if two waves are $y_1 = A\sin(2\pi f_1 t)$ and $y_2 = A\sin(2\pi f_2 t)$, their sum can be rewritten as an amplitude-modulated wave. The envelope of that wave oscillates at $\frac{1}{2}|f_1 - f_2|$, so the loudness — which depends on amplitude squared — waxes and wanes at $|f_1 - f_2|$. If the frequencies differ by more than about 10–15 Hz, the ear begins to perceive them as two separate tones rather than a single wavering note.
If two notes are close but not identical, beats are heard. As the frequencies get closer together, the beat frequency decreases and the pulsation slows down.
When tuning an instrument, the musician’s goal is to make the beats disappear or become so slow that they are imperceptible. That tells the player that the two frequencies are effectively identical. Piano tuners have used this principle for centuries: they strike a reference tuning fork and the piano string simultaneously, then adjust the string’s tension until the wavering sound vanishes.
In an ensemble, players constantly listen for beats between their own instrument and others. By making tiny pitch adjustments — often less than a hertz — they eliminate beats and produce a blended, unified tone. This is especially critical in choirs and string quartets where just a few cents of detuning can create an unpleasant roughness.
Intensity and loudness are not the same thing. Intensity is a physical measure of power per unit area, while loudness is the subjective sensation interpreted by the brain.
Because the ear responds roughly logarithmically, a tenfold increase in intensity is perceived as only about a doubling of loudness. This is why the decibel scale is so useful: it translates physical measurements into numbers that match what we actually experience. A rock concert at 110 dB is not just “a bit louder” than normal conversation at 60 dB; it is $10^5$ times more intense, yet it feels perhaps 16–32 times as loud.
| Intensity ratio | Decibel change | Perceived change |
|---|---|---|
| 1× | 0 dB | Reference level |
| 10× | +10 dB | About 2× as loud |
| 100× | +20 dB | About 4× as loud |
| 1000× | +30 dB | About 8× as loud |
Visual Break — Decision Flowchart
✏️ Worked Examples
Scenario: A listener at an outdoor concert in Sydney moves from 2 m to 6 m from a loudspeaker. Compare the sound intensity at the two positions and explain what this means for hearing exposure.
If the distance only doubled (from 2 m to 4 m), the intensity would become one quarter, not one half. Explain why the inverse-square law means that small distance changes near the source produce large intensity differences.
Scenario: A piano tuner strikes a tuning fork of frequency 256 Hz and a slightly out-of-tune piano string. The tuner hears a beat frequency of 4 Hz. Determine the possible frequencies of the string and describe what the tuner should do next.
If the two frequencies moved closer together, the beats would become slower. If the frequencies matched exactly, the beats would disappear and the sound would become a steady, pure tone.
Scenario: A factory floor measures 85 dB. After installing new machinery, the level rises to 95 dB. Calculate the factor by which the sound intensity has increased.
The level instead rose from 85 dB to 105 dB. What would the intensity ratio be, and why does a 20 dB increase correspond to 100× intensity?
🏃 Activities
A listener stands 4 m from a stage speaker. They then move to 8 m away.
Write one paragraph explaining why a 10 dB increase should never be described as “just 10% more intense.” Use the words logarithmic, intensity ratio, and factor of 10.
A violinist plays a string alongside a 440 Hz tuning fork and hears 3 beats per second.
At a Sydney concert, the sound level is measured as 100 dB at 10 m from the stage speakers.
Earlier you were asked whether 100 dB is only a little more intense than 90 dB.
The full answer: the decibel scale is logarithmic, so a 10 dB increase means 10 times greater intensity, not a 10% increase. This is why decibel changes can carry major real-world consequences for hearing safety.
Now revisit your prediction. What was misleading about treating decibels like a simple linear scale?
Annotate your prediction in your book with what you now understand differently.
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
✅ Check Your Understanding
1. If distance from a point sound source doubles, intensity becomes:
2. A 10 dB increase corresponds to a sound that is:
3. Beats are caused by:
4. Two sounds have frequencies 300 Hz and 306 Hz. The beat frequency is:
5. Beats are heard as a periodic change in:
6. If a musician hears slower beats while tuning, it means the two frequencies are:
7. Explain why sound intensity decreases with distance from a point source. 3 MARKS
8. A listener moves from 3 m to 9 m from a speaker. Compare the intensities at the two positions. 3 MARKS
9. Two notes of 440 Hz and 444 Hz are played together. Calculate the beat frequency and explain how this helps a musician tune an instrument. 4 MARKS
1. C — doubling distance reduces intensity to one quarter.
2. B — +10 dB means 10 times the intensity.
3. A — beats come from superposition of nearby frequencies.
4. D — $|300 - 306| = 6\ \text{Hz}$.
5. C — the loudness rises and falls periodically.
6. B — slower beats mean the frequencies are getting closer.
Q7 (3 marks): As sound spreads away from a point source, the same energy is distributed across a larger and larger area. This means less intensity reaches each unit area as distance increases. That is why sound intensity follows the inverse square law.
Q8 (3 marks): Use $\dfrac{I_1}{I_2} = \dfrac{r_2^2}{r_1^2}$. So $\dfrac{I_1}{I_2} = \dfrac{9^2}{3^2} = \dfrac{81}{9} = 9$. The intensity at 3 m is 9 times the intensity at 9 m.
Q9 (4 marks): $f_{\text{beat}} = |440 - 444| = 4\ \text{Hz}$. So the loudness changes 4 times each second. A musician uses beats when tuning because slower beats mean the played note is getting closer to the reference frequency. When the beats disappear, the notes are effectively matched.
Challenge the boss using your knowledge of sound intensity, the decibel scale and beats. Pool: lessons 1–10.
Tick when you have finished the activities and checked the answers.