When driving my car more than 20 minutes, any rough patches in the road cause something that sounds like it’s under the hood and not far from the steering column to make a knocking sound. Acoustically, it varies from the sound of a diving board vibrating to that of a basketball bouncing perhaps twice in half a second. Occasionally, it makes a high-frequency squeaking sound. My normally reliable mechanic couldn’t figure it out and I’m hoping I can get another mechanic to figure it out (first I need to take them for a drive so they can hear it. Problem is when I stop driving for a few minutes, it takes several minutes of driving for this phenomenon to start up again). Help! 
What engine? Have the motor mounts been checked?
A bouncing engine can cause the exhaust to knock against whatever is nearest. Usually leaves marks.
It’s a 2.0 Liter turbo 4 cylinder engine (NOT the Quatro).
My go-to mechanic, who specializes in Porsche/Audi/BMW couldn’t replicate the noise, but it happens literally every time I drive for more than 20 minutes. Was thinking of visiting a rival mechanic who specializes in German cars. Should I mention what you suggest could be the culprits?
There are a lot of possibilities, but one common cause of rattling is the heat shield on the catalytic converter. And they often only rattle when the converter heats up and expands. These are generally either spot-welded or tack-welded onto the canister’s flanges, and the welds corrode and break. The “fix” is generally to attach the heat shield to the canister with a large worm-drive hose clamp, available for only a few bucks at hardware stores.
Hopefully that’s all it will end up being. That or some other inexpensive clamp or hanger.
Sure, you can suggest things that might be the cause but let the mechanic do his diagnosis. Noises like this are tough to find if you can’t demonstrate it for the mechanic. See if you can arrange to drive the car long enough to make the noise and swing by the shop so your mechanic can hear it. If he is an Audi specialist, I’d give him another crack at it. For tricky things like noises, familiarity with the model and the customer helps.
The turbo 2.0 means you have a larger volume of exhaust “stuff” that can hit things. And as @the_same_mountainbike mentioned, several heat shields.
The “after driving 20 minutes” symptom is a tough one. I would have suggested struts and/or strut mounts if it sounded at all like a small pile of 2x4’s flopping around in the engine compartment. But the “loose lumber” noises usually happen all the time. It still is be something to consider.