what role does this letter with hamza has ؤ? Because this letter as I can hear it's not pronounced so if would it be a difference to write for example instead of فُؤا to write فُءا? Thank you
For a lengthy discussion Hamza, you can search here in this forum search bar for The Story of Hamza - I made 3 posts about Hamza in the beginning, middle and end of the word as Elke said.
In a nutshell, the Hamza on Waw here because the letter before it or the Hamza itself has the vowel (u). Example: فؤاد (fu'ád). The Vowel on (F) is Dhamma (that is -u) and hence the Hamza coming after has to be written on Waw just to make it clear for readers that the letter before has (U) vowel. It is more complex than that but I'm trying to simplify this here. This mainly helps on writing and reading Arabic properly without using diacritics or Harakat to show the vowels on the letters (otherwise typing in Arabic for us would be such a lengthy process).
Just another example of how it works here, take the two words: بئس and بأس. Without any signs for vowels, بئس must be read as (bi's) and بأس as (ba's). The meaning is quite different between the two of course; بئس is a word of damning or cursing and used in the meaning of "misery" of some sort. While بأس means "power" or "hardship".
Bonus: Common phrase in Arabic: لا بأس (lá ba's: alright/it's OK), and also لا بأس عليك (lá ba'sa 3alayk: you are fine/you'll be good).
Maybe this might give you a very first idea about this topic. The main point is here about the Hamza, the glottal stop in the pronunciation. The Hamza can be stand alone but mostly you will find it on a letter. These letters can be Alif, Waw and Ya (here the letters loses his dots below of it). If the Hamza is on one of these letters then you do not pronounce these letters but you have to 'say' an glottal stop in the word, which is also mostly shown with the sign ' before the following vowel. The rules are quiet complex. If you want to know a bit more about the rules just look for TJ's three posts. Hamza at the beginning, at the middle and at the end of a word.
The (hamza) is written on (waw) when it is associated with (Damma) or when the letter before it is - examples : مَسْؤُول (notice the Damma on the hamza) - فُؤَاد (notice the Damma on the letter before hamza) - there is one exception to this rule : when (hamza) is associated with (kasra) or the letter before is - in this case, hamza is written on (nabira) like this : ئُـ or ئِـ
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But what I want to figure out, why the letter 'waw' is muted? If the letter has no influence to the word, why not to write only hamza instead of 'waw' with hamza on it
Because there was Arabic dialects in which hamza was not pronounced - instead of saying ذِئْب people used to say ذيب - and until now in all arabic non standard dialects it's rare when you find hamza in the middle or in the end - you'll find فار instead of فأر
the hamza - fat7a - kasra - tanwiin - shadda - and even the dots were not used in writing - they were added when arabic began to be used by non arabs - this means that the alif - waw - ya2 preceeded hamza in the scriptures - and they don't want to change the scripture by replacing alif waw ya2 by hamza - they choose to add it obove like they do with dots shadda and vowels