In many sentences of Lingvist's Spanish course the Spanish speaker omits sound "s" if it's the last letter of a word and the next word starts with "r". Is this really the way how Spanish speak or it's the defect of sound recording? If it's the real way to speak, then when should I omit "s" and when shouldn't I? And are there any other similar pronunciation rules?
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Regarding your question, the sounds /s/ and /r/ haven't got a clear pronunciation in that specific phonological context. If we want to make a clear distinction between them, we must mark a very subtle pause: los[...]rayos (/los raʝos/). Thus, as a general rule, both sounds are usually softened in such a context (the /s/ sound may even disappear in fast speech).
For further information on how to pronounce /r/ and /rr/, please check the following link.