When I turn off either shower in the house, I hear a high pitched whining noise. It stops if I turn on any other water source in the house, but then starts again when I turn it off. It only lasts about 10 minutes. What is it?

Air in the pipes

Are you on a municipal water system? Do you have a pressure regulating valve? Are you on a well system?

I assume I'm on a municipal system (I don't have a well). I wouldn't know where to look to see if I have a pressure regulating valve... On a friend's advice, I did try turning on every water source in the house at once, starting from the top, but that didn't help.

I would buy a presssure gauge and fittings to attach it to the garden spigot. If your pressure is over 50 psi you need a pressure reducing valve (PRV) to bring it down or you already have a PRV that needs adjustment or replacement. The high pitched whine is temporary excessive pressure blowing past a valve seal, likely the toilet ballcock or frig ice maker.

Look for a PRV valve at the place where you turn off the water to the house. Should look something like this.

Bob
Note that the whining noise is occurring when the shower is turned off and continues for 10 minutes. I'm thinking trash in PRV as it tries to adjust pressure .

I agree, it could be a PRV trying to reduce the pressure, or like I said, maybe another valve like a ballcock or ice maker. Something is venting the excess pressure, your thought about a PRV does make the most sense. Good call Harold.