Sound Intensity and Decibels
Calculate sound intensity and decibel levels for incoherent sources spreading spherically.
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Sound Intensity and Decibels
Concept Overview
Sound intensity (I) is defined as the acoustic power (P) transferred per unit area in a specific direction. For an isotropic source in a free field (where there are no reflections), the power spreads spherically, meaning the intensity decreases with the square of the distance (r) from the source: I = P / (4πr2).
Because human hearing perceives a vast range of intensities, the sound intensity level (β) is commonly expressed in decibels (dB), a logarithmic scale relative to a reference intensity (I0 = 10-12 W/m2): β = 10 log10(I / I0).
Worked Example
Let's calculate the intensity and decibel level for a typical scenario from the module.
- Inputs: Power per source P: 100 mW, Distance r: 5 m, Incoherent source count N: 2
- Convert Power to Watts: P = 100 mW = 0.1 W
- Total Power for N Incoherent Sources: Total P = N × P = 2 × 0.1 W = 0.2 W
- Calculate Intensity I = Total P / (4πr2): I = 0.2 / (4 × π × 52) = 0.2 / (100π) ≈ 6.36619772368e-04 W/m2
- Calculate Decibel Level β = 10 log10(I / I0): β = 10 × log10(6.36619772368e-04 / 10-12) ≈ 88.0388012297 dB
Experiment: Inverse-Square and Multi-Source Addition
Use the interactive Sound Intensity and Decibels module to explore these effects:
- Set the Power per source P to "1" (mW).
- Set the Distance r to "1" (m) and Incoherent source count N to "1". Observe the baseline intensity level (~79.0079 dB).
- Double the distance by setting Distance r to "2". Notice the intensity drops to one quarter, and the level decreases by exactly ~6.02 dB (which is 20 log10(2)).
- Return Distance r to "1", and increase the Incoherent source count N to "10". Observe that 10 times the intensity results in exactly a +10 dB increase in the level (~89.0079 dB).
- Set the Power per source P to "0" (mW) to see the limit case where the logarithmic level drops to "minus infinity".
- Restore Power per source P to 1, Distance r to 1 and Incoherent source count N to 1. The combined and single-source comparison readouts are both shown and match exactly.
- Set Distance r to 0. The real domain error appears and intensity/level readouts disappear. Restore 1, then clear Incoherent source count N, observe its integer error and restore 1.
- Set Power per source P to 1000, Distance r to 20 and Incoherent source count N to 10. The finite intensity is about 0.00198943678865 W/m2 and level about 92.9873014465 dB.
Model Limitations
Ideal Free Field: The drawing is one representative cross-section of a spreading sphere and auto-fits the selected radius; it does not place all sources or predict their individual physical positions. This model assumes a perfectly lossless free field with no reflections, absorption by the air, or obstacles. In real environments, room acoustics (reverberation) drastically alter intensity spreading.
Incoherent Addition: The sources are modeled as mutually incoherent, meaning their intensities add linearly. If sources were coherent, their amplitudes (pressures) would add, causing constructive and destructive interference patterns not simulated here.
Physical vs. Perceived: This calculates the objective physical intensity level (dB). It does not model perceived loudness (Phons), A-weighting, or any biological health recommendations.
References
You can learn more about sound intensity and decibels from this primary reference:
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Sound Intensity and Decibels module.
Try Sound Intensity and Decibels on Riano →