Sound Level With Distance Calculator: Inverse Square Law and Adding Noise

How the noise level drops with distance from a source in the open, and what happens when you add other noise.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Sound spreads out from a small source in the open, so the energy is spread over a bigger area as you move away. Each doubling of the distance lowers the level by about 6 dB. The level at a new distance is the level you measured, minus 20 times the log of the ratio of the distances.

The calculator also turns that around to find the distance at which the level drops to a target. If you give the level of other noise at the point, it adds the two on an energy basis: two equal levels give 3 dB more, and a noise 10 dB below the other adds very little.

This is for a small source with nothing around it. Indoors, near walls or the ground, or for a long source such as a pipe or a road, the level falls more slowly. Measure to confirm any figure that matters.

Formulas used

  • L_2 = L_1 - 20\log_{10}(r_2/r_1)
  • r_t = r_1\,10^{(L_1 - L_t)/20}
  • L = 10\log_{10}\left(10^{L_a/10} + 10^{L_b/10}\right)

Worked example

With the default numbers in the calculator above:

  1. Level at the new distance: 76.94 dB
  2. Distance for the target level: 3.16 m
  3. Drop each time the distance doubles: 6.02 dB

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Level at that point from other noise (0 to skip) 80 dB.

  1. Level at the new distance: 76.94 dB
  2. Combined with the other noise: 81.74 dB
  3. Distance for the target level: 3.16 m
  4. Drop each time the distance doubles: 6.02 dB

Common questions

Why does 6 dB drop for each doubling of distance?

Because the area the sound spreads over grows with the square of the distance, so the energy per area falls to a quarter, which is 6 dB.

Why do two 80 dB noises make 83 dB, not 160?

Decibels are on a log scale. Adding equal energy doubles it, which is 3 dB more.

Is this the same as noise exposure?

No. Exposure depends on time as well. Use the noise exposure calculator for the 8-hour level and dose.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Managing noise and preventing hearing loss at work Code of Practice (Queensland); standard acoustics texts on free-field spreading; AS/NZS 1269 (licensed).

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