The exposure clock protects one worker over one shift. A site also has neighbours, and they are protected by a different instrument: a twenty-four-hour average with the night made to count for more. It is the metric North American land-use, airport and highway rules are written in, and many municipal bylaws borrow it.

The day–night average sound level: $L_{dn} = 10\log_{10}\!\left(\dfrac{15\cdot 10^{L_d/10} + 9\cdot 10^{(L_n+10)/10}}{24}\right)$. Read aloud: L-d-n equals ten log of, fifteen times ten to the L-d over ten, plus nine times ten to the L-n plus ten over ten, all over twenty-four. $L_d$ is the daytime average level in dBA, the energy average over the fifteen hours from 07:00 to 22:00. $L_n$ is the night-time average level in dBA, over the nine hours from 22:00 to 07:00. $L_{dn}$ is the single figure a rule is written against, and it is usually what you solve for. The 15, the 9 and the 24 are hours.

Take it apart and there is nothing new in it. Each level goes back up into an energy, exactly as in Adding noise. The hours weight the energies. One logarithm is taken at the end. The only addition is the +10 riding on $L_n$: every night hour is entered ten decibels louder than the meter read it, which is ten times the energy.

That penalty is policy, not physics. No meter hears a night decibel differently. It is a 1974 US EPA judgement about sleep, carried into housing, aviation and transit practice, and the arithmetic around it is the ordinary energy average. Two benchmarks ride with it: 55 dBA is the level identified as protective outdoors, and 65 dBA is the long-standing line above which housing is treated as incompatible.

One fact is worth memorising. A night exactly 10 dB quieter than the day is neutral: the penalty cancels it, every hour carries the same weighted energy, and $L_{dn} = L_d$. A 60 dBA day with a 50 dBA night is 60.0. Hold the night at 60 as well and the figure climbs to 66.4: six and a half decibels added by nine hours, purely because they are night hours. So the first move on any problem is to add ten to the night and see which period now leads.

Read backwards, the relation is a budget. The cap fixes the total weighted energy, one period spends part of it, and the other gets the rest: $L_n = 10\log_{10}\!\left(\dfrac{24\cdot 10^{L_{dn}/10} - 15\cdot 10^{L_d/10}}{9}\right) - 10$. The trailing $-10$ turns the penalised night back into a level a meter would read, and it is the step people drop.

The named mistake is forgetting the night penalty. Leave it out and you have computed the plain 24-hour average, a real quantity with a different name, and it under-reports every site that works after 22:00. And the nugget for the field: because a night hour is charged tenfold, moving the same plant from the night shift to the day shift can cut its contribution by close to 10 dB. That is often a cheaper mitigation than any barrier, and it costs nothing but a schedule.