Reb Gil writes here about noise during davening - noise from talking, and noise from other people who are davening.
The former type of noise is not of great interest to me; I think the primary reason for talking during davening is not sinfulness but rather thoughtless synagogue design, cramming people close to each other for 90 to 150 minutes during which they are supposed to be unnaturally focussed only on prayer and not at all on each other.
The latter, though - noisy davening - is something I’ve been contemplating lately.
The gemara actually presents four different reasons to recite Shemoneh Esreih silently:
1) Chanah prayed silently (Berachot 31a)
2) Silent prayer shows faith in Divine omniscience (Berachot 24b)
3) Loud prayer would embarrass sinners who wished to admit their sins (Sotah 32b)
4) Loud prayer would distract others (Berachot 24b)
It is also noteworthy that the Aruch haShulchan (Orach Chaim 101:8) adds a fifth factor: A hushed demeanor is appropriately respectful when standing before royalty.
(Note that there are also reasons to daven aloud; see Biur haGra Orach Chaim 101 להגביה and Aruch haShulchan Orach Chaim 101:7-8.)
But let’s look at this on a more abstract level. Certainly, silence is of practical value - סיג לחכמה שתיקה as well as נצור לשונך מרע - but silence is also an expression of spiritual depth.
Consider Avraham’s silence at the instruction to slaughter his son Yitzchak, and the accolades he earns for his unquestioning loyalty.
Consider Aharon’s silence at the death of Nadav and Avihu, and the praise heaped upon him for his articulate inexpression.
Consider Chanah’s silent plea for a child. Despite the fact that the gemara reads all manner of imprecation into her non-words, the plain presentation is of a servant of Gd, nobly stoic in her suffering.
Consider the Davidic counsel of דום לה' והתחולל לו, Pray silently for Gd and tremble (Ibn Ezra's translation) for Him .
And, finally but most significantly, consider the contrast between Eliyahu’s thundering rage and the Divine קול דממה דקה with which Gd rebukes him, a sound so thin as to be nearly inaudible, and yet deep enough to contain the majesty of the Creator of All.
This is a silence of presence, of pent-up power, of tzimtzum, of a Being who surely can thunder like Eliyahu but who chooses the containment of Chanah.
To me, the silence of Shemoneh Esreih is an attempt to capture this noble state of expressive restraint.
And yet, the Rambam (Hilchot Tefilah 5:9) and Shulchan Aruch (Orach Chaim 101:2-3) do not choose this route, but instead opt for the prosaic, if socially praiseworthy, concern of distracting others. Oh, well. There's a lesson in that, too.