Well, you've got a bunch of current going through a coil of wire inside the circuit breaker.

You'll remember from third grade science class that, if you take a coil of wire and run current through it, it makes a magnet.

This is AC, so the magnetism is reversing polarity 60 times a second...

So some metal part inside the circuit breaker is therefore MOVING 60 times a second, which makes a nice humming noise.

It's normal. Pushmatic breakers AREN'T quite normal nowadays (I don't think they've been manufactured in the last 30 years) but the humming noise is, in and of itself, normal.

The "pushmatic" breakers are the type used in Federal Pacific panels which many electricians refer to as "widow makers". Federal Pacific panels and breakers are no longer made as the company lost their UL rating many years ago. Suggest you do a google on FP. These panels were notorious for causing fires as many times the breakers would not trip on an overload. Further suggest you look into replacing this panel. By any chance do you have aluminum wiring?

The house is a small (1100sf) single story "all electric" ranch style vacation home built in 1970. It has baseboard electric heat, electric hot water heater and an electric stove. The panel box is an ITE brand with a 200 amp main breaker. The wiring in the house is copper. There is 12 -15 breakers in the box. They are all pushmatic style. Many of them are 240V because of the electric baseboard heater circuits. What concerned me about the humming noise is I don't remember hearing it when the electric hot water heater was wired directly. I only noticed it after I installed the clock timer switch in the circuit. It may have been there before and I just didn't hear it. I also don't hear humming coming from any of the other breakers. That's what made me wonder if I should replace this breaker or if the time clock had anything to do with the noise.

... Pushmatic were from ITE. The Federal Pacific breakers were "Stab Lok".

Pushmatic breakers, as I recall, aren't much more reliable than Federal Pacific, but they're not one and the same.

You will only hear the humming when the water heater is heating. ie, if the water is already hot, there will be no noise.

More than likely you just never happened to have your ear to the panel while the water heater was heating. The timer shouldn't have anything to do with it.

Of course, while you were installing the timer, the water cooled off, and thus the heater started heating as soon as you turned the power back on.

Go around your house and crank all of the electric heaters up to 90°, then listen to your panel. I bet you'll get a buzz.

Just do it quickly, or your house will get real hot.

It's only humming because it forgot the words.

But seriously, ac power hums, especially in the circumstances outlined.