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This is what I was thinking too. Sometimes an exhaust restriction sounds exactly like what he's describing.Hard to believe that this could be it but maybe a partially clogged catalytic converter. Where is the noise coming from? Front, middle, back? Maybe have an experienced friend be outside and accel hard bast him/her and see if they can pinpoint area where noise is coming from. Maybe get back pressure test on exhaust. Can you open hood and do the hard accel and hear it? Does the noise increase with speed and/or intensity thru the acceleration process?