Waiting 40 or 80 days

This is incorrect, and it is a very common misconception.

To understand the source of this idea, it is important to distinguish between Torah law and the halachot as currently practiced.

According to Torah law (VaYikra 12:1-8), there are three distinct time periods post-partum. The following is a description of the law as it was once followed, not as it applies today.

• During the first week from childbirth, for a boy, or the first fourteen days, for a girl, a woman was a yoledet. The restrictions that applied were identical to those of a niddah.
• Over the next thirty-three days, for a boy, or the next sixty-six days, for a girl, any bleeding was considered dam tohar, a unique form of uterine bleeding that did not render a woman niddah.
• Any bleeding after day forty (7 + 33) or eighty (14 + 66) from childbirth rendered a woman niddah.

The halacha as currently practiced is as follows:

As always, a woman is rendered a yoledet by the childbirth (she also becomes niddah during labor or with uterine bleeding following a cesarean). If she is able to get a good hefsek taharah and count shivah neki’im, she may immerse in the mikveh as soon as the initial time period that the Torah stipulates for a yoledet – seven or fourteen days – has passed.

However, based on many halachic developments over the years, a woman is now rendered niddah by uterine bleeding in the dam tohar period.

Thus, in practice, the process of becoming tehorah after childbirth is identical to that following a menstrual period. A woman waits until bleeding has stopped, performs the hefsek taharah, counts shivah neki’im, and goes to mikveh. (The only difference is that she cannot immerse until at least fourteen days after a girl is born; however, it is very rare for bleeding to stop soon enough for this to be possible.)

Some communities developed the custom of delaying immersion until the end of forty or eighty days.  Many halachic authorities condemned any such stringency and today it is not practiced.