i know what a relay does, and how to wire them, but that isn't the problem. He's not getting a thump when turning on/off radio, only when closing the doors. However, he does mention that 20 seconds after the car is off it will stop, so maybe? Always a possibility i guess. if you wanted to wire a relay you could connect 85 - radio accessory, 86 - ground, 30 - amp turn on wire (cut), 87 - 12v. This would turn the amp on/off along with the radio, eliminating the delay...
i'm curious if you're getting bass notes with the car running, while playing music? I would slam the door and check if it's still happening then.