Acoustics & Noise formulas — 11 solvers

Allowable Noise Exposure Time

T = \frac{8}{2^{(L - L_c)/q}}

Acoustics & NoiseHow long a person may be exposed to a steady noise level before reaching a full daily dose. The answer depends entirely on which exchange rate the jurisdiction uses, and the two in common use disagree by hours.

Combining Sound Levels

L_t = 10\log_{10}\!\left(10^{L_1/10} + 10^{L_2/10}\right)

Acoustics & NoiseDecibels do not add. Two 60 dB machines running together measure 63 dB, not 120 — the energies add and the logarithm is taken again at the end. This is the most misused arithmetic in noise control, and the fix is one line long.

Composite Transmission Loss

TL_c = 10\log_{10}\!\left(\frac{S_w + S_d}{S_w\,10^{-TL_w/10} + S_d\,10^{-TL_d/10}}\right)

Acoustics & NoiseA wall with a door or a window in it, rated as one partition. The two paths add by AREA AND TRANSMISSION, never by decibels, and the answer is the most sobering arithmetic in building acoustics: the weak element sets the rating almost by itself.

Distance Attenuation from a Point Source

L_2 = L_1 - 20\log_{10}\!\left(\frac{r_2}{r_1}\right)

Acoustics & NoiseMove twice as far from a point source in the open and the level drops 6 dB — every time, regardless of the starting level. Ten times the distance is 20 dB. This is the cheapest noise control there is, when there is room for it.

Eyring Reverberation Time

T_{60} = \frac{0.161\,V}{-S\,\ln(1-\bar{\alpha})}

Acoustics & NoiseSabine's equation, corrected for rooms that actually absorb. Once a room is treated, sound is lost on every reflection rather than continuously, and the logarithm in the denominator is what accounts for it — Sabine's form runs long in exactly the rooms people pay to have treated.

Mass Law Transmission Loss

TL = 20\log_{10}\!\left(\frac{\pi m f}{\rho_0 c}\right) - 5

Acoustics & NoiseFor a single limp panel, blocking sound is almost entirely about weight: transmission loss climbs about 6 dB every time the surface density doubles, and another 6 dB every time the frequency doubles. It is the reason there is no light way to stop bass.

Noise Reduction Coefficient (NRC)

\mathrm{NRC} = \frac{\alpha_{250} + \alpha_{500} + \alpha_{1000} + \alpha_{2000}}{4}

Acoustics & NoiseThe single number on an acoustic panel's data sheet: the plain average of its absorption coefficients in the 250, 500, 1000 and 2000 Hz octave bands, rounded to the nearest 0.05. Convenient, widely quoted, and it hides everything that happens below 250 Hz.

Sabine Reverberation Time (RT60)

T_{60} = \frac{0.161\,V}{A}

Acoustics & NoiseHow long a sound takes to fade by 60 decibels after the source stops — the single number that decides whether a room is a concert hall, a classroom or a swimming pool. Big rooms ring; absorptive rooms do not.

Sound Power Level to Sound Pressure Level

L_p = L_W + 10\log_{10}\!\left(\frac{Q}{4\pi r^{2}}\right)

Acoustics & NoiseA machine's sound POWER is a property of the machine; its sound PRESSURE is what a meter reads at a given place. This converts one to the other in a free field, using the distance and the directivity of wherever the machine is sitting.

Sound Transmission Loss

TL = 10\log_{10}\!\left(\frac{I_i}{I_t}\right)

Acoustics & NoiseHow much of the sound striking a partition never gets out the other side, in decibels. A 40 dB wall lets one ten-thousandth of the incident energy through — which still sounds like something, because hearing is logarithmic too.

Total Absorption (Sabins)

A = S_1\alpha_1 + S_2\alpha_2 + S_3\alpha_3

Acoustics & NoiseAdd up the room: every surface contributes its area multiplied by how much of the sound striking it never comes back. The total, in metric sabins, is the A that Sabine's reverberation equation divides into the volume.