Whether kicking the tail pipe reproduces the noise or not, I am pretty sure that you are merely hearing the effects of a loose or bent heat shield on the exhaust system. This situation is something that is not consistent, and can change with both engine speed and the operating temperature of the exhaust system.
The only way to tell for sure is to put the car up on a lift in order to examine the exhaust system while the engine is running. A helpful mechanic can usually remedy this problem with a clamp applied in just the right place. Do NOT allow anyone to remove the heat shield. If it wasn’t necessary, the manufacturer would not have spent the money to install it in the first place.