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I'm planning to go to shooting ranges soon, and am looking for ear protection.

Let's say that I get an earmuff with a noise rating reduction (NRR) of $X\ \textrm{dB}$ and earplugs with $Y\ \textrm{dB}$. When one wears both earplugs and earmuffs (i.e. "doubling up"), what would be the final noise rating reduction in terms of $X$ and $Y$ (i.e. an equation), provided that one properly wears them?

Here are some numerical examples:

  1. Peltor Sport Tactical 500 Active Earmuffs (NRR $26\ \textrm{dB}$)

  2. Decibullz Earplugs (NRR $31\ \textrm{dB}$)

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  • $\begingroup$ Based on my experience, it's a good idea to wear ear plugs and ear muffs. While people on this site can calculate the noise reduction that you would get, it just makes common sense to protect your hearing as much as possible. $\endgroup$
    – David White
    Commented Jul 11 at 1:58
  • $\begingroup$ @DavidWhite, if that were true, the only sensible thing to do would be to spend our lives in individual acoustic isolation chambers. Fact is, even on a shooting range, (especially on a shooting range?) it's good to be able to hear what is happening around you. $\endgroup$
    – Solomon Slow
    Commented Jul 11 at 3:42
  • $\begingroup$ David White, true that. Solomon Slow, that's why there is active ear protection. $\endgroup$
    – Superman
    Commented Jul 11 at 4:32
  • $\begingroup$ @SolomonSlow I shoot at an indoor gun range. Some patrons shoot AR-15 pistols, which are equivalent to AR-15 rifles with a hand grip instead of a stock. Such weapons are so loud that you can feel the concussion several stalls away from the shooter. In that environment, you don't want to hear what is going on around you. $\endgroup$
    – David White
    Commented Jul 11 at 14:23
  • $\begingroup$ @DavidWhite, I guess that depends on what you mean by "hear." When I said, "hear," I didn't mean experience the concussion in all it's glory,* I meant, hear things well enough ("things" including commands spoken by the range safety officer) to be aware of the situation around you. [*IDK about AR-15s specifically, but I've been on an indoor range where somebody was sighting in his deer rifle, so I guess I've got a rough idea.] $\endgroup$
    – Solomon Slow
    Commented Jul 11 at 14:34

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If filter #1 reduces the sound level by $X$ decibels, and filter #2 reduces it by $Y$ decibels, the overall noise would be reduced by $X + Y$ dB.

Mathematically, a reduction of $X$ decibels in the sound level corresponds to a reduction in the sound intensity (sound energy per area per time) by a factor of $10^{X/10}$. This means that if a sound wave passed through your earmuffs and then through your earplugs, its intensity would first be reduced by a factor of $10^{X/10}$ and then by a factor of $10^{Y/10}$. The overall reduction would therefore be $10^{X/10} \times 10^{Y/10} = 10^{(X+Y)/10}$.

But this would be equivalent to passing the sound through a single device that reduced the sound intensity by $X + Y$ decibels. In other words, the decibel reductions of the two filters can simply be added together to get the total decibel reduction.

That said, from the document linked in the comments, it appears that NRR ratings are not calculated in so straightforward a way (I'm a bit surprised by this, to be honest.) So the NRRs cannot simply be added.

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  • $\begingroup$ Based on this document (osha.oregon.gov/OSHAPubs/factsheets/fs80.pdf), we can only add $5\ \textrm{dB}$ to the higher of the NRR rating, so the equation provided in the answer isn't correct. $\endgroup$
    – Superman
    Commented Jul 10 at 18:20
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    $\begingroup$ Hmm, interesting. I'm pretty sure that the answer I gave is correct for the question "if a sound wave is reduced by $X$ decibels and then by $Y$ decibels, what is the total reduction of the sound wave in decibels?" But it appears from your document that NRRs are not related to broad-spectrum sound intensity reductions in the same straightforward way. I'll edit my answer to reflect this, and I'll encourage you to not accept my answer and wait until a better one comes along. $\endgroup$
    – Michael Seifert
    Commented Jul 10 at 18:29
  • $\begingroup$ I found out that the $5\ \textrm{dB}$ rule of thumb mainly comes from an empirical study: audiologyonline.com/articles/… $\endgroup$
    – Superman
    Commented Jul 13 at 2:33
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    $\begingroup$ I think the reason you can't just add $X$ and $Y$ is because the mechanisms for attenuation are not independent of each other. If you have over-ear protection, a major source of sound that gets through would be through the skull, and adding in-ear protection would not help with that issue. I doubt if an analytical analysis of the full problem of adding protection has ever been done, and so the empirical approach is probably the best. $\endgroup$
    – Michael M
    Commented Jul 17 at 15:58

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