To me, Wave failed because it wasn't a product. It was an engineering exercise. There was a lot of interesting technology in Wave (eg the Wave protocol itself, federated servers).

But what problem did it solve?

No one really seemed able to answer that.

When I looked at Wave I saw a set of primitives from which you could build email, IM, Twitter or any other number of communication mediums. The problem is that people could already do all of those things elsewhere. Did Wave make it easier? If it did, it might've stood a fighting chance. But it didn't. The UI was complicated. On most browsers it was slow.

Part of what made Google the search behemoth that it is (apart from the obvious algorithmic improvements over what came before it) was the simplicity of it. It was the simplest of pages (compared the shotgun portals of Yahoo and the like) with a giant text box. Anyone could look at it and know what to do. What's more it was fast.

So I would take Wave as an object lesson in what happens when you seek to solve an interesting engineering problem rather than build a product.