Will they send offer letter within 15th April
There is certainly a possibility that you will get moved from the wait list to acceptance before April 15th, but there is absolutely no guarantee or requirement of further contact. By "possibility", I mean this happens to students every year and it isn't particularly unusual - but I also mean that this only happens to a minority of students, and is hard to predict due to the nature of statistics and small numbers per program. Some programs might admit 2 students and both admitted students accept, while others might admit 20 yet go 2+ full rounds of wait list contacts before admission decisions are officially finished.

The general advice in the US is that you are safe to wait until close to that April 15th deadline for acceptance if you are not certain that's where you want to go, even if you are just holding out to see if any wait listed programs might accept you. Waiting past that deadline is risky unless you request and obtain an extension, as programs are well within their rights to rescind their offers and move on to their own wait list after the dead line.