OK, downloading now.
Now, I'm on illustrator 10, but something odd is happening. I click the central shape of the logo (which remains white in my version of ill.) and it selects that shape, BUT it also selects a large frame (may be coloured white, and grouped with other objects, why is this?) No amount of ungrouping solves this.
Closely looking at that middle shape, it's still not a closed shape. Take a look at some screens I've taken, where I separated the pieces.
http://img183.imageshack.us/my.php?image=picture1li5.png
See? These parts should all be one continuous, and connected line. Otherwise, you simply cannot fill it, much less use a gradient. It's not the same as paint programs, where you can break down the line into pieces and still have it register as an outline for a fill.
I downloaded a web image of the logo, and traced that part myself. Have a look at this image to see where my points are, and see that it's connected.
This one shows the lower portion, and what I was referring to when I said I'd use 4 points to create the wave.
http://img406.imageshack.us/my.php?image=picture2kc7.png
The middle section. Remember, longer anchor points give smoother and more natural curves. This is a great shape to practice that.
http://img406.imageshack.us/my.php?image=picture3os4.png
Remember, this is just how I would do this, so I'm not trying to say this is the correct way. Others will do it differently I'm sure. This is just designed to be an example so you can see how closed paths look, and a guide to see how to space out and plan your points.
Just try to keep your points simple. Space them out, and draw out the anchor points. Don't switch to other tools WHILE creating the path, because the last created point will become a corner point, and you don't want that. To close off the shape, click once again on the first point (the mouse cursor should alter, to have a small circle by it)
Sorry about that links, photobucket was down for maintenance, so I had to use imageshack, ugh.