It does, it also looks like a similar view painted by Alan Lee, so he must have based his on this one by Tolkien.
You're right, the movie Frodo missed those chances to show the steel within his character. It came out a little at the council though, when he volunteered to bear the ring "though I do not know the way." Very brave.
Dimly Frodo saw the river below him rise, and down along its course there came a plumed calvalry of waves. White flames seemed to Frodo to flicker on their crests and he half fancied that he saw amid the water white riders upon white horses with frothing manes.I remember (and liked) the white horses but I don't remember seeing white riders. Does anyone remember seeing white riders in that scene?