You may have observed that most speakers of Hindi pronounce ऋ as essentially रि [rɪ] thus pronouncing words like ऋषभ or कृषि or आकृति as रिषभ, क्रिषि and आक्रिति respectively while speakers of Marathi and Gujarati pronounce it as रु [rʊ] thus saying रुषभ, क्रुषि and आक्रुती. It is a completely different matter that most Hindi speakers do not pronounce ष as a distinct sound. They use the same sound for ष and श, roughly the voiceless post-alveolar fricative, which also explains why most of us spent a lot of sweat and blood in memorizing when a word had श and when it had ष—there were no phonetic clues to differentiate between what was essentially a spelling distinction.
But I digress. Let’s get back to the slippery ऋ business. Quite a few Marathi speakers believe that their pronunciation is more superior or more-correct, so to speak, compared to the Hindi counterpart. While a community can choose whatever pronunciation it wants for a character, out of academic interest, I tried to investigate the situation.
Let’s try to look at what ऋ really is. ऋ is usually called a syllabic consonant, meaning that it acts as the nucleus of a syllable. To understand it better, look at the word *button* (click here and listen to the pronunciation). You’d notice that there is no audible vowel sound after [t]. This doesn’t work if you pronounce it with an Indian accent, so check out the pronunciation I have given the link to. Despite there being no vowel sound after [t], you get the feeling that the word has two syllables. That’s because the [n] sound here acts like a syllabic consonant, that is, it is the nucleus of the second syllable [tn]. We usually associate syllables with vowels, so it seems weird to see a consonant making a syllable, but it’s perfectly normal—syllables are usually interpreted based on peaks of sonority, and sufficiently sonorous sounds like [m, n, r, l] can also act as syllable nuclei. Look at the pronunciation of the word bottle, for instance. Here [l] is acting like a syllabic consonant.
Let’s put this into perspective. ऋ is supposed to be a syllabic consonant, so it won’t have a vowel with it and, going by the definition, that rules out both variants [rʊ, rɪ]. As a community, speakers of Marathi have ended up choosing one, while Hindi speakers have chosen another.
What does ऋ sound like? I have never heard Vedic Sanskrit spoken by a native speaker, but thankfully, we have modern languages that routinely use syllabic consonants. Czech has a tongue twister Strč prst skrz krk that has no vowels—just four syllables with r in a syllabic role. Click here for the pronunciation of this sentence by a native speaker. Here’s another video of a guy explaining the tongue twister.
What’s the bottom line? That it’s perfectly OK to call ऋ either रु or रि; while one of the two might sound slightly closer to the actual ऋ, neither pronunciation probably matches it exactly. It does’t matter either way. And there is really no point in calling one pronunciation superior to the other.