Separate spherical spreading from background intensity
An ideal isotropic source distributes acoustic power over a sphere of area 4πr², so its source-only intensity is P/(4πr²). A separate uniform incoherent background adds intensity, giving I(r)=S/r²+B when S is the source intensity at one metre. This sum assumes the sources are uncorrelated; coherent waves must instead have their pressure amplitudes and phases combined.
Learning outcome: Fit source strength and background from the first two radii, freeze the third-radius prediction and only then reveal it. Repeat with zero background and compare level drops. Propose one fresh observation that could detect a directional rather than isotropic source.
Starting recap
Separate local oscillation from propagation, amplitude from its square, and a calibration fit from a fresh prediction. Use declared phase, medium frame and boundary conditions.
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Try
For S=3 µW/m² and B=1 µW/m², readings are 4 and 1.75 µW/m². Their difference 2.25 recovers S=3 and B=1. The fresh 3 m prediction is 4/3 µW/m², and isotropic power is 12π µW. The level difference is 10 log10(1.75/4)≈−3.59022 dB, not −6.02 dB.
Open the experimentPredict
Why is the total intensity drop from 1 to 2 m smaller than a factor of four when the background is positive?
Experiment
Fit source strength and background from the first two radii, freeze the third-radius prediction and only then reveal it. Repeat with zero background and compare level drops. Propose one fresh observation that could detect a directional rather than isotropic source.
Separate spherical spreading from an incoherent background
Worked teaching model · synthetic values · Isotropic stationary point source, free field without absorption/reflections, independent spatially uniform incoherent background. Indications at radii 1 and 2 m calibrate source power and background; radius 3 m is fresh validation. Level uses reference 1e−12 W/m², not a health or perceived-loudness judgment.
Reset keeps your prediction, captures and practice history.
Inverse-square signal with background
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: prediction from inferred source/background (solid).2: fresh record at 3 m (dashed). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Inverse-square signal with background
prediction from inferred source/background
| radius (m) | total intensity (µW/m²) |
|---|---|
| 1 | 4 |
| 1.0375 | 3.78705 |
| 1.075 | 3.596 |
| 1.1125 | 3.42394 |
| 1.15 | 3.26843 |
| 1.1875 | 3.12742 |
| 1.225 | 2.99917 |
| 1.2625 | 2.88217 |
| 1.3 | 2.77515 |
| 1.3375 | 2.677 |
| 1.375 | 2.58678 |
| 1.4125 | 2.50364 |
| 1.45 | 2.42687 |
| 1.4875 | 2.35584 |
| 1.525 | 2.28998 |
| 1.5625 | 2.2288 |
| 1.6 | 2.17188 |
| 1.6375 | 2.11882 |
| 1.675 | 2.06928 |
| 1.7125 | 2.02296 |
| 1.75 | 1.97959 |
| 1.7875 | 1.93892 |
| 1.825 | 1.90073 |
| 1.8625 | 1.86483 |
| 1.9 | 1.83102 |
| 1.9375 | 1.79917 |
| 1.975 | 1.76911 |
| 2.0125 | 1.74071 |
| 2.05 | 1.71386 |
| 2.0875 | 1.68844 |
| 2.125 | 1.66436 |
| 2.1625 | 1.64152 |
| 2.2 | 1.61983 |
| 2.2375 | 1.59923 |
| 2.275 | 1.57964 |
| 2.3125 | 1.56099 |
| 2.35 | 1.54323 |
| 2.3875 | 1.5263 |
| 2.425 | 1.51015 |
| 2.4625 | 1.49473 |
| 2.5 | 1.48 |
| 2.5375 | 1.46592 |
| 2.575 | 1.45245 |
| 2.6125 | 1.43955 |
| 2.65 | 1.4272 |
| 2.6875 | 1.41536 |
| 2.725 | 1.40401 |
| 2.7625 | 1.39311 |
| 2.8 | 1.38265 |
| 2.8375 | 1.37261 |
| 2.875 | 1.36295 |
| 2.9125 | 1.35366 |
| 2.95 | 1.34473 |
| 2.9875 | 1.33613 |
| 3.025 | 1.32785 |
| 3.0625 | 1.31987 |
| 3.1 | 1.31217 |
| 3.1375 | 1.30476 |
| 3.175 | 1.2976 |
| 3.2125 | 1.29069 |
| 3.25 | 1.28402 |
| 3.2875 | 1.27758 |
| 3.325 | 1.27136 |
| 3.3625 | 1.26534 |
| 3.4 | 1.25952 |
| 3.4375 | 1.25388 |
| 3.475 | 1.24843 |
| 3.5125 | 1.24316 |
| 3.55 | 1.23805 |
| 3.5875 | 1.2331 |
| 3.625 | 1.2283 |
| 3.6625 | 1.22365 |
| 3.7 | 1.21914 |
| 3.7375 | 1.21476 |
| 3.775 | 1.21052 |
| 3.8125 | 1.2064 |
| 3.85 | 1.2024 |
| 3.8875 | 1.19851 |
| 3.925 | 1.19473 |
| 3.9625 | 1.19107 |
| 4 | 1.1875 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
fresh record at 3 m
| radius (m) | total intensity (µW/m²) |
|---|---|
| 3 | 1.33333 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Intensity at one metre | 4 | µW/m² |
| Intensity at two metres | 1.75 | µW/m² |
| Inferred source-only intensity at one metre | 3 | µW/m² |
| Inferred incoherent background | 1 | µW/m² |
| Inferred isotropic acoustic power | 0.0000376991 | W |
| Third-radius prediction from two-point fit | 1.33333 | µW/m² |
| Fresh third-radius record | 1.33333 | µW/m² |
| First-radius intensity level | 66.0206 | dB re 1e−12 W/m² |
| Second-minus-first intensity level | -3.59022 | dB |
Subtract the background before testing inverse-square spreading; logarithmic levels do not add linearly.
I(r)=S/r²+B gives S=4(I1−I2)/3 and B=I1−S for radii 1 and 2 m. Isotropic power is 4pi S after converting micro-units. Independent incoherent intensities add; coherent pressures must be combined before squaring. A constant background makes the measured level drop less than 6.0206 dB when distance doubles. A modeled level is not a safety limit, a hearing test or a promise about subjective loudness.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Subtract the background before testing inverse-square spreading; logarithmic levels do not add linearly.
I(r)=S/r²+B gives S=4(I1−I2)/3 and B=I1−S for radii 1 and 2 m. Isotropic power is 4pi S after converting micro-units. Independent incoherent intensities add; coherent pressures must be combined before squaring. A constant background makes the measured level drop less than 6.0206 dB when distance doubles. A modeled level is not a safety limit, a hearing test or a promise about subjective loudness.
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