After each of the makkos, Moshe Rabbeinu davened to Hashem for it to stop. After the makkah of frogs, the pasuk says, “…vayitzak Moshe el Hashem al devar hatzefardi’im asher sam l’pharaoh – and Moshe cried out to Hashem regarding the frogs that he inflicted on Pharaoh” (Shemos 8:8). This is the only makkah where we find that the Torah uses the word “vayitzak [cried out]” in reference to how Moshe davened to Hashem. By the other makkos, the Torah uses the word “vayetar.” The Sifsei Chachamim asks the question that he says was bothering many people: why does the Torah change the wording for the makkah of the frogs?
The Sifsei Chachamim answers that the frogs were croaking and making a lot of noise. The halacha (Shulchan Aruch Orach Chaim 101:2) states that one who is davening must recite the words loud enough for him to hear what he is saying. Since the frogs were making loud noises, Moshe could not hear his own words. Therefore, he had to scream his tefillah in order to hear his own words.
This opinion is not universally accepted, as we will see.
The Netziv, in his sefer Hemek She’eilah (143:6), points out a contradiction between a Tosefta and a Yerushalmi. The Tosefta (Berachos 3:9) says that one should not say the words loud enough to hear them himself. The Yerushalmi (Berachos 2:4) says that optimally one must recite the words of davening loud enough to hear them himself. The Netziv suggests that there is no dispute between the Tosefta and the Yerushalmi. Rather, the Tosefta agrees that one must say the words loud enough to hear them himself. The Tosefta, however, is referring to a situation where there is a lot of noise and he cannot hear himself – even if he speaks at the decibel in which he would hear himself if there was no noise around him. In this situation the Tosefta says that one should not daven louder in order to hear himself over the noise; rather, it is sufficient to say the words at a decibel whereby he would hear himself if it was not noisy.
The Netziv seemingly directly contradicts Sifsei Chachamim, namely that in a noisy place one must recite the words of davening louder so he can actually hear what he is saying. The Netziv believes that the halacha that one must hear what he is saying is not to be taken literally; rather, the halacha only requires that one recite the words at a certain decibel that could generally be heard – regardless of whether he can actually hear himself.
The Shulchan Aruch Orach Chaim 101:2 states that one must not daven in his heart alone, rather he must “cut” the words with his lips and allow himself to hear the words with his ears in a whisper, while not allowing his voice to be heard. If one is alone where he will not disturb others and he cannot have the adequate kavana while only whispering the words, he may daven out loud. However, if he is with others and davening out loud will disturb their davening, one may not daven out loud.
The Mishnah Berurah explains that one who davens out loud is demonstrating that he is of small emunah, since he is implying that Hashem cannot hear his quiet tefillah. Additionally, one who raises his voice while davening is compared to those who worship idols, for that is how they would serve them.
In summation there are two reasons why one should never daven aloud, which can be suspended if one needs to concentrate, and another issue of distracting others, which is not permitted under any circumstances.
Based on the ruling of the Mishnah Berurah I would conclude that if one cannot hear his own words due to loud noises around him, he should still not raise his voice. Only for the exception of kavana is one permitted to do so in private. However, davening in a loud place would still be in violation of the two reasons one should not daven aloud; it shows that Hashem cannot hear a silent tefilla and resembles the practice of idol worshipers.