The Gemara doesn’t consider soothing sounds Hashmo’as Kol in Hilchos Shabbos. That is not the greatest Raaya, but the simple Svara is that way, anyway. It might be called Shiras Habria but it’s not actual music.
If soothing, natural sounds would be a problem it would be Assur to stroll by a brook. You do find an Issur of Tiyul, strolling, in Hilchos Tish’a B’av, but I don’t know about Hilchos Availus. Regardless, a recording of a walk is not a walk.