We've just started a fresh campaign with both experienced and new players alike. During our second session one of the new players was browsing their spells during combat and I saw confusion, then contemplation, and then they pulled out a massive grin and started googling.

On their turn, they turned to me and asked: "So Thaumaturgy says one of its effects is that 'Your voice booms three times as loud as normal for 1 minute' right? A normal human speaking is 60db, would that mean that if I shout I can break 180db and damage their hearing to do damage?"

As the session was nearly over I told them to hold the thought for the rest of the encounter so we can figure it out afterwards just to keep the combat going quicker for the group https://mobdro****/ https://vidmate****/.

This was four days ago now. I still have no idea how to run it. Its a creative AOE damage solution and I don't want to damage that new player spark, however its also quite strong effect for a cantrip and I don't know where to begin for damage.

TLDR; My player wants to use thaumaturgy's increased voice volume to Canary their enemies to death and I don't know how to run it.

Any help or ideas would be great!

I'd say "It's a game and that particular game element does not specify any resulting damage to the affected target and even if the description could be interpreted from some basis in reality, everyone in the area, including yourself, would have to make a saving throw which the spell does not specify." I'd say if they want to push it then the affected targets would have an intense ringing in their ears making vocal communication difficult for 1 minute, including the caster.

Compare it to an actual sound-based damage cantrip, thunderclap. Whatever you decide, it should be weaker than that. I would say it could cause “pain,” but not actual damage, or at most 1 damage in a very small area, perhaps a 10’ cone or something at the most, so it would have to be in real close. It’s enough to reward the ingenuity, but not so much it unbalances anything.

The simple rules based answer is that the spell description doesn’t say it does damage, so it does not do damage.  You can do differently if you desire, of course. If you want to allow for damage, look at other attack cantrips for guidelines on damage amounts. 

All the above, also, you can science right back at them. Decibels are a logarithmic scale, where a change in power by a factor of 10 corresponds to a 10 dB change in level. I can't trivially figure out how "three times louder" directly translates to the power (I'm too lazy, for one thing), but they aren't getting anywhere near what they want.

To expand on what jl8e is saying:

Decibels are logarithmic so multiplying a given DB results in addition on a log scale. This site shows a good graph of what is happening: https://energyeducation.ca/encyclopedia/Logarithmic_scale

For a given value, multiplying its intensity by 10 results in something 10 db higher. Something 3x louder is ~5 dB higher. So 60dB x3 = 65 dB.