Sound / Np → B
Convert Nepers of pressure ratio to Bels relative to reference pressure
Convert RMS sound pressure and logarithmic pressure levels with an explicit reference. Convert Nepers of pressure ratio (Np) to Bels relative to reference pressure (B). Find the formula, worked examples, unit definitions, and practical guidance below.
Reverse: Bels relative to reference pressure to Nepers of pressure ratio →
How to convert nepers of pressure ratio to bels relative to reference pressure
Default reference pressure is 20 µPa, the conventional air SPL reference. Values are unweighted RMS pressure levels, not dBA or sound intensity. Logarithmic levels may be negative; physical pressure must be positive. Changing the reference changes the reported level.
Use x for the value in Np, b for the intermediate value in Pa RMS, and y for the result in B.
b = p₀ × 10^(x ÷ 2.30258509299); y = 2 × log₁₀(b ÷ p₀)
Here p₀ = 0.00002 Pa. Static examples and tables use that default reference. The calculator supports another positive reference pressure.
Worked examples
10 Np ≈ 8.68588963807 B
Step 1: p₀ × 10^(10 ÷ 2.30258509299) ≈ 0.440529315896 Pa RMS.
Step 2: 2 × log₁₀(0.440529315896 ÷ p₀) ≈ 8.68588963807 B.20 Np ≈ 17.3717792761 B
Step 1: p₀ × 10^(20 ÷ 2.30258509299) ≈ 9703.3039082 Pa RMS.
Step 2: 2 × log₁₀(9703.3039082 ÷ p₀) ≈ 17.3717792761 B.
Results are rounded to 12 significant digits. Converting units does not increase the precision of the original measurement. Very large or small values outside JavaScript’s finite numeric range are rejected.
About Nepers of pressure ratio
Nepers of pressure ratio expresses a pressure ratio logarithmically: level = 2.30258509299 × log₁₀(p/p₀). Here p is RMS pressure in pascals and p₀ is the selected reference pressure. Zero level means pressure equal to the reference, not silence.
About Bels relative to reference pressure
Bels relative to reference pressure expresses a pressure ratio logarithmically: level = 2 × log₁₀(p/p₀). Here p is RMS pressure in pascals and p₀ is the selected reference pressure. Zero level means pressure equal to the reference, not silence.
Definitions and sources
Carry the same physical assumptions into subsequent physics calculations.
Nepers of pressure ratio to bels relative to reference pressure reference table
| Nepers of pressure ratio (Np) | Bels relative to reference pressure (B) |
|---|---|
| 1 | 0.868588963807 |
| 2 | 1.73717792761 |
| 5 | 4.34294481903 |
| 10 | 8.68588963807 |
| 20 | 17.3717792761 |
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