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So you were being vectored for the final approach course (ILS/RNAV/whatever) and the controller forgot about you for some reason (busy). Assume you called them and they were not replying. Also assume you were not yet cleared for the approach but they told you to assume the x approach. Would you blow through the final approach course and wait for them, even though you might conflict with traffic, or turn inbound even though you had not been cleared and potentially take a beating later?

Yes, blow right through it. You should never deviate from a clearance to "take a beating" later simply because you think you know better than the controller. If the controller is busy it is for a reason, and it is very possible they are extending you on purpose for spacing. You could very well conflict with traffic even more by taking your own turn. It is far better to let them fix the overshoot they caused than to battle the FAA over a flight violation.

If you genuinely think the controller may have forgotten about you, you should continue to attempt contact, while pressing IDENT with each transmission. If it is so busy that you can't get a word in edgewise, press IDENT anyway. I have done this and received an immediate turn without even having to transmit on the radio. It is designed to get their attention, just don't overuse it.

The only exceptions where you should deviate from an assigned vector are if you are lost comm or have an emergency. In either case make sure you are squawking appropriately.

I would consider a vector towards terrain a pending emergency if not turned within some appropriate period of time. When I was instructing in the Navy, during simulator rides I would force a missed approach at McChord AFB, give a climb to 5000' and a left turn towards Mt Rainier. I would then fail the crew's radios. It was always a good lesson in terrain awareness, especially for young aviators from flat states.

It's an oddball situation because normally once you are put on a heading that will intercept final, the approach clearance is included with the heading instruction ("XXX, turn right heading YYY, cleared for the approach").

But lets say the controller actually just gave you a heading to final and nothing else, and you never heard from them again. The question becomes, is this a valid comm failure or not.

If it's a real comm failure, and you continue through the final approach course and just continue off to who knows where, what is the controller to do for separation of traffic? You are now like a horse that escaped the barn, and the controller now has no idea of where you are going next.

Assuming all efforts to raise ATC fail, you go to the basic IFR comm failure procedure and proceed logically based on the comm failure protocol; clearance, expected clearance, and all that. In this case it's pretty clear to expect the approach you are being vectored to.

So if it was me, I would be frantically trying to make contact but once I was convinced I had a real comm failure I'd select 7600 and just fly the approach.

Once ATC sees the 7600 selection, that's what the controller will expect me to do and will be able to adjust traffic flow to suit. That last thing to do in a comm failure situation in a terminal area is to head off in unexpected directions.