Notes & formulas

1) Energetic addition

What it is. Sum of independent sound levels by adding energies (not dB directly).

Formula. \( L_{\mathrm{add}} = 10\log_{10}\!\big(\sum_i 10^{L_i/10}\big) \)

Tip. The result is slightly above the largest input; two equal sources add ≈ \(+3.0\ \mathrm{dB}\).

2) Energetic subtraction

What it is. Remove a known component from a total level (e.g., subtract background).

Condition. Defined only if \( L_{\mathrm{comp}} < L_{\mathrm{tot}} \). Formulas. \( E_{\mathrm{remain}} = 10^{L_{\mathrm{tot}}/10} – 10^{L_{\mathrm{comp}}/10} \),
\( L_{\mathrm{remain}} = 10\log_{10}(E_{\mathrm{remain}}) \)

3) Energetic average (reported in dB)

What it is. Logarithmic mean: average energies, then convert back to dB.

Formulas. \( E_{\mathrm{avg}} = \frac{1}{n}\sum_i 10^{L_i/10} \),
\( L_{\mathrm{avg}} = 10\log_{10}(E_{\mathrm{avg}}) \)

Note. Do not take an arithmetic mean of dB; dB are logarithmic. Energy-averaging preserves the physics.

4) Frequency weighting (A/C/Z)

What it is. Convert between dB(Z), dB(A) and dB(C) at the 1/3-octave center frequency \(f\) using tabulated corrections \(A(f)\) and \(C(f)\).

Conversions.
From Z: \( A = Z + A(f) \), \( C = Z + C(f) \).
From A: \( Z = A – A(f) \), \( C = Z + C(f) \).
From C: \( Z = C – C(f) \), \( A = Z + A(f) \).

Interpretation. A ≈ human sensitivity at moderate levels (attenuates lows); C is flatter at low frequencies; Z is flat (instrument bandwidth).

Notes common to all. \(E\) denotes an energy-equivalent linear quantity (e.g., power or mean-square pressure). Numeric outputs use a fixed one-decimal format.