Low rumbling noise at cold start

So for the last week my Civic has been making a low, rumbling sound (almost like an overly powerful bass system or a muffler) and vibrating whenever the car is idling from a cold start. The noise is most pronounced when the car is started but continues for a short time while I am driving, but only when I take my foot off of the gas in low RPM (below 1.0k). Whenever I accelerate, the noise stops, but when I go in reverse the noise is becomes louder. Also, the noise is very faint from outside of the car, but very pronounced from inside. After about 20-30 minutes of driving it usually goes away. What could be causing this?

Have someone check the flex pipe under the engine. This flex pipe allows the engine to move under torque without effecting the rest of the exhaust system.

Tester

If is it the flex pipe, will it damage my car if I leave it unrepaired over time?

It may leak exhaust gases and that’s not good for you. Like tester says: Get it checked out to make sure.

It depends where the flex pipe is leaking as runs under the oil pan.

But normally the flex pipe will rot out where it will require replacement. Here’s an example of the flex pipe.http://www.jcwhitney.com/garage-pro-oe-replacement-down-pipe/p3085779.jcwx

Tester