Oh, I hate it that a part of life that we always expect to be happy - married to our lover and having children together--is anything but happy for you right now. I'm sure it feels the same for your wife.
The first thing I want to tell you is that postpartum depression is more common with the second child, and it often shows up as anger instead of tears, although there may be tears and sadness, too. And there may be anxiety. Sometimes we women who have just had a baby think we can't do it, we've ruined our lives, we'll mess up, etc.
The first thing to investigate is whether your wife is depressed.
It was awful. I couldn't talk about it because I was afraid I'd come across as not loving my children. It's a horrible black cloud to live under, but is also, luckily, a very easily treated illness. I could not believe anything could help me at that time. But a little pill rapidly changed everything, and counseling made it even better.
Don't suffer and don't let your wife keep feeling awful. Get to a doctor or psychiatrist and have her checked for something that is a purely physical condition for which there is wonderful treatment.