Does the buzzing sound increase / decrease with engine speed / stepping on the throttle?

If it does then it might be something being rotated by the belt that comes off the crank shaft: alternator, air condition compressor, tension pulley, anti-freeze pump, or the belt itself. In the old days, we used to call this the "fan" belt but the cooling fans are powered by electricity and not the belt anymore.

From past experience, the bearing on the tension pulley tends to dry out / make noise first on all sorts of vehicles I've had. After running the engine for a while / short trip, the belt and bearings in the tension pulley and cir conditioning compressor clutch are warm. I "turn off the engine" and then squirt some WD40 on both sides of the tension pulley bearing and leave it soak in the next 15 mins or more - more doesn't hurt anything. I know it is supposed to be sealed but I'm sure that after time there is some gap that allows water into the bearing and that the heat of the engine and inside the bearing dries out the grease. A little WD40 wets the old grease and allows it to move around and do a better job of lubricating inside there. See if that helps.

If the belt, itself, is old / stiff / cracking, it could be slipping / vibrating upon initial start up and then as it gets warm from operating it get more supple / grips and quiets down. They do sell spray cans of "Belt Dressing" at car part stores but it is a temporary fix. When a belts starts acting like that and it shows signs of wear, it is better to change it than have it break at an inconvenient time.

Worse / higher costs are the anti-freeze water pump and/or the air conditioner clutch. You can buy remanufactured water pumps cheap but the labor is high because it is a pain to do. The A/C clutch is just expensive.

Let us know what it is.

 

 

what it is, is your bendix gear inside your starter is failing. The noise will happen only on cold mornings. When it begins to happen twice a week, I'd suggest getting a new starter. All us Toyota/Honda owners are familiar with this problem. Seems to happen to Denso starters more for some reason. I've replaced the starter with the same noise in my Vibe and my Fit. No big deal; it is just your starter's time to fail.

 

If the noise occurs with the function of the starter, I agree with BurntZ. Sorry for going down the wrong rabbit hole.

 

If it is a starter motor failing to engage, replace starter sooner than later. Once upon a time, I let a problem like this go too long that it destroyed the teeth on flex plate. Flex plate was attached to the torque converter so it became quite an expensive repair.

It was on a 1985 Honda Accord. They had Mitsubishi or Mitsuba starters back then.

 

Is your car manual trans?