2001 Honda Civic Charging problem

I replaced the alternator on my 2001 Civic EX a year ago and just got a new battery, and the belts and connections seem to be in order. When I start the car after it's been sitting for several hours, the battery light flashes for about 10 minutes and electrical components are not at full strength -- the headlights and dashboard lights are clearly dimmer than normal. But after that 10-minute period, the battery light stops flashing and the headlights return to full strength and stay at that level as long as the car is on. And if the car has only been off for a brief period -- 30 minutes or so -- it starts without incident. The alternator seems to be working at least part of the time, why is it or the charging system not working when I first start my car?