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I found this page with good illustrations of how to pronounce each sounds in AE.

http://www.uiowa.edu/~acadtech/phonetics/english/frameset.html
Is the /r/ sound featured on this page what is called an alveolar approximant by phoneticians?
http://en.wikipedia.org/wiki/Alveolar_approximant
Unfortunately, I'm only able to produce the retroflex approximant

http://en.wikipedia.org/wiki/Retroflex_approximant
I'm I right to think that the former sound is the more prevalent one? Unfortunately, I've not been able to get this sound down yet and am only able to produce some sort of hissing sound when I try. The instructions on Wikipedia on how to pronounce the sound are pretty vague, so I would be grateful if someone gave me more detailed instructions on it.
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I think you should ignore the 2 Wikipedia pages and stick to the first link.

I am unaware of the terms Alveolar or Retroflex approximants. Forget those and focus on the speaking. What you should do is hear the sound you want to make, in this case it is an "r", and then think of sounds that require you to pronounce the "r" sound.

For example: immitate a lion's roar, or a dog's bark (roof, roof).

Another strategy is to start by humming with your mouth closed, but hum so that you are making a sound as loud as you would speak. Then, open your mouth and try to make the sound.
iLrrr-n I'm I right to think that the former sound is the more prevalent one?
Oops, what I meant to say is of course Am I right to think that the former sound is the more prevalent one? I would have edited the post, but didn’t find that feature anywhere.

To the point. The problem isn’t producing an R-sound, I can produce the retroflex approximant easily and well, but it seems impossible to produce the r-sound demonstrated here on the page of the University of Iowa:

http://www.uiowa.edu/~acadtech/phonetics/english/frameset.html

manner -> liquid -> /r/
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Liquids do have allophones: one of the liquid /r/ allophone is retroflex. Many times, learners just focus on some abstract sound without looking at the environment. For instance, AmE /t/ has 8 allophones.

Even if a foregin language has the AmE /t/ phoneme, it may not have the same set of allophones that /t/ have in AmE.
iLrrr-nThe problem isn’t producing an R-sound, I can produce the retroflex approximant easily and well, but it seems impossible to produce the r-sound demonstrated here on the page of the University of Iowa:

http://www.uiowa.edu/~acadtech/phonetics/english/frameset.html

manner -> liquid -> /r/
The R sound presented there is indeed the most prevalent one in American English. It may seem impossible, but millions of us do it effortlessly hundreds of times a day. If you're getting hissing, you're not voicing it and you may be holding your tongue too close to your palate.

CJ
So did I get it right? The center and the front of the tongue is raised towards the hard palete while the tip of the tongue remains relatively stationary, the sides of the tongue are pressed against the upper teeth to prevent any lateral air flow and there is no tongue curling? I think I might be getting the hang of it, but it's really hard to keep the tongue from curling back at least a little.

How about the usage? Is this /r/ used in all situations, by the people who use it, or are some other allophones used in certain situations?
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iLrrr-n,

If you are just interested in producing american R, here is a tip from "American spoken english in Real life: fast natural, urgent survival foreign accent begone!" by DG Davies.

" r + vowel = ur + vowel. First say u. Tongue back into mouth does not move while lips smile changing to next vowel.
write - right urait, three thuri, brow buraun, etc."

Here is a video that follows the above advice:
uKDKH257j18


Note the rounded ness of /r/
I found a very nice quiz about the letter R in British English. Quiz doesn't sound very promising but the article explains how and when to pronounce British R. Then you can test your knowledge in multiple choice questions and the good thing is that the answers are explained.

This is the link

http://www.squidoo.com/quiz-when-to-pronounce-the-letter-r-in-british-english