It's Physics time!

Ok, what is "loudness"? To say it easy, the ammount of noise something produces is the amplitude of a compression wave of air. When it hits a surface, it transmits through that material and/or is reflected by it. The transmitting sound is distorted in the process before exiting on the other side.

Now, we measure sound pressure - which correlates with the ampliude of the compression wave - in decibel, or dB for short. dB is a logarithmic scale, where 3 dB means 2 times the energy and \$\sqrt 2\$ the amplitude of the wave. In other words: something that is 10 times as loud has 100 times the energy and is 20 dB louder.

It also has a neat chart that tells us how many dB something creates when it is a set distance away on [wikipedia][1]. Combat could from own experience (a sword clashing onto a helmet next to me) reach the noise of a vuvuzela horn at 1 meter distance (120 dB) but usually will be a bit lower than what a Jackhammer does at the same distance (100 dB).

Dish out the math already!

So, let's assume at 1 meter (\$r_1 = 1\text{m}\$) it's as loud as a noisy road in 10 meters distance, so \$p_1 = 90\text{dB}\$.

What is that pressure at the door? \$p_2 = p_1\frac{r_1}{r_2}\$ tells us. let's assume the room has 2 meters to the door. So \$p_2=90\text{dB}\times\frac{1\text{m}}{2\text{m}}=45\text{dB}\$. 45 dB is already the noise level of a normal conversation.

Let's assume the door does not distort the sound or reduce the sound pressure in a meaningful way (it does, but one can't calculate that easily at all). So we have 45dB of noise to be heard in 20 feet (=6.096m). That's \$p_3=45\text{dB}\frac{1}{6.096}=7.38\text{dB}\$, so less than the noise of a light breathing or leafs rustling.

A curtain (which again should dampen the sound but does not for the sake of the experiment), and another 10 feet (=3.048m) later, the sound pressure is down to \$p_4=2.42\text{dB}\$. Remember, a normal conversation is 40 to 60dB.

But what if they are louder?

If we assume the people yell and scream at the top of their lungs and produce 120 dB at the 1m Range, the sound travels still 11.144 meters, so the sound pressure drops - assuming no doors and curtains but one straight line without obstacles - \$p_2^*=120\text{dB}\frac{1}{11.144}=10.77\text{dB}\$, and taking each of the obstacles as a new source for the sound, just about 3.22dB. Inaudiably under a normal conversation or noisy mess hall.

That doesn't make sense to me!

Why does the formula seem to quirk out for iterative setup in comparison to cummulative? The cummulative assumes we have one corridor of sound. The iterative assumes we transmit the sound through some perfect, non dampening material between three areas of a set length. Think like a tube in which you put in thin layers of foil: even their mere presence does dampen the sound.

Ok, but what if somebody yells from the door?

Ok, let's assume a Goblin makes it to the door and yells out of it at the strongest he can muster, the 120 meters above. Down the hallway at the curtain that is still 22.14dB, and in the canteen a mere 7.26dB - or 13.12dB assuming it is just one corridor. Again, ignoring the dampening properties of the curtain material.

tl;dr:

Nope, at this distance the sounds will be drowned out by the noise of the mess hall.

That doesn't however stop a sloppy goblin that is late for lunch from passing the door that the players fight behind and hearing a particular loud scream (ca. 60dB without door dampening, a guesstimated 30 to 50dB, depending on door material and thickness, on the other side) and then run to alert the rest.

In that case the closed door doesn't help the players at all, as their own combat noise would drown out these alerted goblins advancing down the corridor and taking position on both sides of the door. Running in armor should be roughly 70dB at 1 meter. Assuming the heroes are 2 meters in the room and the goblins rub their bellies against the door in going into position, this is 35dB at the player's positions and that surely will drown out in their own combat noise. But again, the door and air in the hallway dampens that sound before pressing through the door, so... if you rule that a goblin manages to alert the garrison, and they manage to be quick enough to come while still fighting, the players should not hear them coming.

^{Disclaimer: The calculation makes a few assumptions and simplifications and takes a few shortcuts that result in the numbers being unrepresentative for real physics calculations. It does however allow an estimation as the results should be within the same order of magnitude.} [1]: https://en.wikipedia.org/wiki/Sound_pressure#Examples_of_sound_pressure