Asynchronous or Synchronous
One-to-one or one-to-many

For example, I'd put the telegraph in the asynchronous one-to-many or one-to-one bucket(s). Email? That's asynchronous 1-to-1 or 1-to-many. Voicemail? That's asynchronous voice, generally 1-to-1. IM? Synchronous 1-to-1. I won't go on at length. I agree with your core point that most new communication tools do not always have obvious use cases up front.

My issue, though, is that I can't think of many successful communication / collaboration tools that didn't set out to solve some easy to describe problem. Two examples:

Voicemail – I wanted to talk to you but you didn't pick up. Here's what I wanted to tell you
IM – Rather than phone or email you, I'd like to chat in real-time

So, I have a hard time bucketing Google Wave in my simple 2×2 system. What i fear, too, is that most uber-collaboration suites I've seen have failed. And I think that's because most people are reasonably facile managing different communication tools for different use cases. An uber tool that only excels by unifying might not capture the imagination of the average user. It's clear to me that Google Wave is solving some high-end or specific use cases for enterprises.

Is your assertion that Google Wave is trying to make email better? Gmail does that. Is it supposed to make collaboration broadly more useful?

What is the use case for email? telephone? writing letters? posting to forums? subscribing to mailing lists? chatting online? sharing status updates a la Twitter?

Communication.

Wave is just another form of communication, and its use cases are as diverse as there are people to use it.

As for what problems is it trying to solve, here's a few:

Email has many copies of a conversation – Wave has one hosted, shared copy.
Email threads with branches are hard to follow and reconstruct. Wave handles this is one place.
It's hard to add someone into an email thread and share the history of the discussion – this is easy in Wave.
Emails cannot be edited after sending. Waves can.
It's hard to respond to specific points in an email thread. In a wave you can reply at a particular point easily.

There are many more which have been written about too.

I agree Wave is unfamiliar and its not obvious how to use it, but it's no different than when any other new form of communication came out. When mobiles came out with SMS, not everyone had one, and people wondered what use text messaging was. Now look at all the use cases that have evolved.

It would also help adoption once they manage to allow sending and recieving using 'standard' email adresses rather than username@googlewave.com

Thanks so much for the detailed reply. I really enjoyed reading it. My real concern is that Wave might be overkill. Why create an entirely new product when adding more collaboration tools (particularly some asynchronous ones) to the existing suite of tools in GDocs might have delivered more immediate value to users?

I work on an internal collaboration team of a huge company and we are trying to get people to use a mish-mash of all our tools together, but after getting into Google Wave I realized this would be a dream come true for most use cases that I'm trying to sell to people.

I honestly think that Wave is really just Google's Business Collaboration suite, or at least that is where it would provide the most value, as people are forced to work on things together all the time, garnering tons of emails all with chained replies (see comments below in red/green/blue), as well with the meeting minutes and just generating docs.

I think it's also hard to get a good feel for Wave's usefulness with such a limited amount of people on it, and it's really just not going to make sense for most people, but for some tasks it's just right, it's like a wiki page, but chat at the same time, like a live brainstorming or mindmapping solution. The problem is that it's good a specific situations, not everything, and most people don't know what situations they are good for, and even less ever need to use those situations in their personal lives.

Anyways, just my thoughts so far, it's still seems really buggy and unpolished, and most people are so disappointed because it's not primarily a personal tool, it's just the hottest toy people want.