My version:

Reference images:

Analysis (whoo, scientific): A lot of Arwen and Eowyn's styles are so simple as to not require analysis, so I'll lump a lot of them together. This one, however, is technically two styles. Picture 3 is actually the first style that she's wearing as she rides up; the costume is slightly different as well. The hairstyle appears to be the same as in the Departure From Rivendell style, so check that page for directions. The style she wears as she walks up to Frodo is show in pictures 2 and 3. The hair is worn down and wavy, but it's not parted. This style is only shown for a minute or two, then the hair and costume switch to the Flight to the Ford outfit. So I guess that would make it three hairstyles...

Directions: I'm only going to give directions for the second style here. It's the basic Arwen style shown in quite a few scenes throughout the trilogy. Don't part your hair, but just let it fall naturally down your back. If your hair isn't naturally wavy, you can get a similar look by braiding your hair while it's wet, then letting it dry and unbraiding it. If you do this, make sure you either braid all the way down to the ends or curl the ends a bit, otherwise you end up with wavy hair most of the way down and stick-straight hair for the last couple inches. That's it!