Power steering valve

This is the sound.

I've been researching this online for the last 4.0 hrs and have discovered something called a restrictor on this attached picture. There is also one present on the replacement, albeit only some of them. This is what was rusted out on my old line. I am not sure it goes all the way thru the line, perhaps only on the outside of it. So I am planning on adding this new line in place of the custom line I had made.

I also discovered another restrictor in the coolant return line. I will put the old restrictor in the line I made or buy the hose as shown on the ecs pic. Will let you know what and how it resolves the whining noise. One right up talks about the continued cavitation so I think it's moving through there too fast. Will let you know the outcome so you can share in future as needed.

Epilogue : I tried to replace the custom pressure hose I had manufactured with the factory replacement which had the factory installed restrictor which was missing from my custom build. After 4.0 hours of trying to install the Gates brand hose, it was off enough within the metal curly bends and entry bend that it was off by 1/4" and would not align with the bolt orifice that could be forced by the left hand only while holding on to the banjo bolt trying to screw it on with index finger only. So, after 4 hours, and about 40 tries I abandoned that and tried going back to my custom hose shop with the factory hose to see if they could install a restrictor. They could, and installed one inside the hose.
I asked what size he used and he said it was quite small. I was able to install the custom line, which re-used the original Audi pipe bends and angles and was successful with the dry fit and a one handed install, and a lot of prayer. I then installed the new return cooler hose with its restrictor. After I got the car front off the ground and performed the bleed procedure, along with replacing the power hose banjo bolt with a new one which had a spring and plastic bearing inside, theorizing it was worn as well, it had no whining on start up, nor during the test ride. A very slight normal whine can be detected but I would consider that normal. So the very loud whining in my case was missing restrictors in the replacement lines, which when resolved, solved the problem. Thanks for the intelligent discussions !

One more thing. I was able to purchase a brand new power steering pump from Europa Parts for $185.00 that was less than the Bosch re-manufactured from Rock Auto. It was made by the same manufacturer ZF that made the original for Audi. It was quieter and it's so nice to have a new pump in there.