Addition of sound waves

This experiment allows us to discover that the addition of two identical sound sources does not lead to a doubling of the perceived intensity but to a lower gain. It introduces the notion of logarithmic decibel scale.

If one loudspeaker produces sound at 60 decibels, how loud are two identical loudspeakers playing together? Most people intuitively answer 120 decibels, doubling the perceived loudness. The actual answer, approximately 63 decibels, reveals a fundamental and often counterintuitive aspect of acoustics: the decibel scale is logarithmic, not linear. This mathematical property reflects how our ears perceive sound intensity, compressing an enormous dynamic range into a manageable scale. The human ear can detect sounds ranging from a whisper at about 20 dB to a jet engine at 140 dB, a ratio of sound intensities spanning 10 billion to one. Understanding the logarithmic nature of decibels is essential in audio engineering, environmental noise assessment, and architectural acoustics. This experiment provides students with hands-on experience of this principle by measuring the combined sound level of two independent noise sources and discovering that doubling the number of sources produces only a modest 3 dB increase.

Activity overview:

The student uses three smartphones: one to measure the sound level and two others to emit white noise. After calibrating each transmitter smartphone to 60 dB, the student measures the sound level when the two sources transmit simultaneously. The experiment is repeated with different types of sounds to analyze whether the sound increase remains constant and to understand the differences observed.

Level:

Medium and High School

FizziQ

Author:

Duration (minutes) :

30

What students will do :

- Measure and compare the sound level of one source versus two identical sources playing simultaneously
- Discover that combining two identical sound sources produces an increase of approximately 3 dB, not a doubling
- Understand the logarithmic nature of the decibel scale
- Investigate whether the type of sound (pure tone vs. white noise) affects the result
- Apply the formula L = 10 × log(I/I₀) to predict the combined sound level

Scientific concepts:

- Superposition of waves, Logarithmic scale of decibels, Difference between pure tones and complex noises, Constructive and destructive interference, Sound power and acoustic intensity

Sensors:

- Microphone (sound level meter in dB)

Material needed:

- Three smartphones with the FizziQ application, A quiet space with few echoes, A tape measure to measure distances

Experimental procedure:

Open FizziQ on three smartphones. Designate one as the measuring device and the other two as emitters.
On the measuring smartphone, open the Sound Level (dB) sensor and verify it reads the ambient noise level (ideally below 40 dB).
On emitter smartphone 1, use the FizziQ synthesizer to produce white noise at a moderate volume. Adjust until the measuring phone reads approximately 60 dB at a distance of 30 cm.
On emitter smartphone 2, produce the same white noise at the same volume. Individually verify that it also reads approximately 60 dB at 30 cm.
Place both emitter smartphones equidistant from the measuring phone (about 30 cm), side by side.
First, record the sound level with only emitter 1 playing for 10 seconds. Note the average value.
Stop emitter 1. Record with only emitter 2 playing for 10 seconds. Note the average value.
Now play both emitters simultaneously for 10 seconds. Record the combined sound level.
Compare the three measurements. The combined level should be approximately 3 dB higher than each individual source.
Repeat the experiment using a pure 440 Hz tone instead of white noise on both emitters. Note any differences in the result.
Try a third trial with different frequencies on each emitter (e.g., 440 Hz and 880 Hz) and compare the results.
Record all data in your FizziQ notebook and calculate the theoretical prediction using L_total = L₁ + 10 × log(2) ≈ L₁ + 3 dB.

Expected results:

When each source individually produces about 60 dB and both play simultaneously, the combined level should measure approximately 63 dB (±1-2 dB), confirming the 3 dB rule for adding uncorrelated noise sources. White noise sources should follow this rule reliably because they are uncorrelated. For pure tones of the same frequency, results may vary depending on the phase relationship: if the waves happen to be in phase, the increase could reach up to 6 dB; if partially out of phase, the increase may be less than 3 dB. For pure tones of different frequencies (440 Hz and 880 Hz), the 3 dB rule should hold well since the signals are uncorrelated. Measurement precision is typically ±1-2 dB due to environmental noise and smartphone microphone calibration differences.

Scientific questions:

- Why does doubling the number of sound sources increase the level by only 3 dB instead of doubling it?
- What is the physical meaning of the logarithmic scale in the context of human hearing?
- Why might two pure tones of the same frequency produce a different result than two white noise sources?
- If you combined 10 identical sound sources, what would the expected increase in decibels be?
- How does this principle apply to real-world situations like traffic noise or a crowd cheering?
- What is the relationship between sound intensity (in watts per square meter) and sound level (in decibels)?

Scientific explanations:

When two sound sources of the same intensity are combined, the increase in sound level is not 100% but approximately 3 decibels. This counterintuitive property is explained by the logarithmic nature of the decibel scale.

The sound level in decibels is calculated according to the formula: L = 10 × log(I/I₀), where I is the measured sound intensity and I₀ is the reference intensity. When two identical, uncorrelated sounds are added together, their intensities (not their amplitudes) are added.

Thus, doubling the sound intensity corresponds to an increase of 3 dB. Behavior may vary depending on the nature of the sounds.

For white noise (containing all frequencies at equal intensity), the addition generally follows this 3 dB rule. On the other hand, for pure sounds of the same frequency, interference phenomena can occur: if the waves are in phase, the increase can reach 6 dB (constructive interference), while waves in opposition to phase can cancel each other out (destructive interference).

Extension activities:

Frequently asked questions:

Q: The combined sound level is much higher or lower than 3 dB above each individual source. What went wrong?
R: Ensure both sources are producing the same volume individually (within 1 dB of each other). If using pure tones of the same frequency, phase interference may cause unexpected results; try white noise instead. Also verify that background noise is well below the source level.

Q: Why do you recommend using white noise instead of a pure tone for this experiment?
R: White noise contains all frequencies at random phases, making it statistically uncorrelated between the two sources. This ensures the intensity addition rule applies cleanly. Pure tones can create phase-dependent interference effects that complicate the measurement.

Q: Is the 3 dB rule always exact?
R: The 3 dB rule applies exactly for two identical uncorrelated sources. In practice, small differences in source levels, reflections, and measurement uncertainty typically produce results in the range of 2-4 dB increase.

Q: Can I hear the 3 dB difference?
R: A 3 dB change is generally considered the smallest difference that is perceptible to the human ear under controlled conditions. It corresponds to doubling the sound power, though it does not sound twice as loud. A perceived doubling of loudness requires approximately a 10 dB increase.