What you are hearing is a combination of sounds that come directly from the roof (caused by rain) and sound that is reflected off the walls, ceiling and floor.

The reflected sound (aka echo) is often much more irritating than the direct sound.

You can "deaden" the floor with carpet and treat a wall with wall hangings, perhaps even erect some bookshelves.

You only need to treat one of a pair of opposite walls so the task is not as difficult as it might sound.