Composite Transmission Loss

This is the most useful arithmetic in building acoustics, and it produces the result people refuse to believe. Sound does not average across a partition — it goes preferentially through whatever resists it least, so the elements must be combined on TRANSMISSION, weighted by area, and only converted back to decibels at the very end. Convert each element's TL to its transmission coefficient $\tau = 10^{-TL/10}$, form the area-weighted sum $\sum S_i\tau_i$, divide the total area by it, and take the logarithm once.

Work an example and the lesson lands. An 18 m² wall rated 45 dB with a 2 m² door rated 20 dB in it — 10 % of the area — comes out at 29.9 dB. Not 45, not the area-weighted 42.5 that averaging decibels would suggest, but a shade under 30. The door carries 97 % of the transmitted energy. Upgrade the wall to 55 dB and the assembly moves to about 30.0 dB: ten decibels of wall bought a tenth of a decibel of result.

The extreme case is worth carrying around as a rule of thumb. An unsealed hole transmits everything, $\tau = 1$, so a gap of 1 % of the area caps the composite at 20 dB and a gap of 0.1 % caps it at 30 dB, no matter what surrounds them. This is the arithmetic reason an undercut door, an unsealed service penetration, a back-to-back electrical box or an open transfer grille destroys a partition, and it is why airtightness and acoustic sealant are not finishing details but the substance of the work. If you can feel air moving through it, sound is moving through it.

Two working consequences. First, always fix the weakest element before improving anything else — the composite can never be better than the worst path, and money spent elsewhere while a weak path remains is money spent for a fraction of a decibel. Second, the same equation is the budgeting tool: fix the target, fix the wall, and solve for how much glazing or door the assembly can carry. The answer is generally smaller than the architect hoped, and it is far cheaper to say so at the drawing stage than after the complaint.