"...the pure sound of merriment..."
There are some things in the movies that seems less "hokey" - and more true to the book - when I look at them in screencaps instead of just rushing by at full speed. This exchange of laughter between Gandalf and Frodo is one of them. This is truly an example of laughter being "the pure sound of merriment" as Sam heard it, rather than the response to something humorous:
But Sam lay back, and stared with open mouth, and for a moment, between bewilderment and great joy, he could not answer. At last he gasped: 'Gandalf! I thought you were dead! But then I thought I was dead myself. Is everything sad going to come untrue? What's happened to the world?'
'A great Shadow has departed,' said Gandalf, and then he laughed, and the sound was like music or like water in a parched land; and as he listened the thought came to Sam that he had not heard laughter, the pure sound of merriment, for days upon days without count. It fell upon his ears like the echo of all the joys he had ever known. But he himself burst into tears. Then, as a sweet rain will pass down a wind of spring and the sun will shine out the clearer, his tears ceased, and his laughter welled up...'
In fact, my clearest memory of this kind of laughter was in response to the same event. I was sitting in our living room while reading "The White Rider" for the first time, and when Gandalf appeared - alive - I was so happy I laughed out loud. My younger brother was walking through the room and asked, "What's so funny?" I said, "Nothing's funny. I'm just so happy!" He shook his head and kept going. (To put this in context, I was 15/16 at the time, which means he was 12/13 - I'm pretty sure he'd understand it now.)
In this scene, movie-Frodo plays the part of book-Sam in some ways: the moment of being caught "between bewilderment and joy" that's shown on the previous page, and the response of laughter. I have mixed feelings about this decision. I've always felt that Tolkien carefully avoided showing us the reunion between Frodo and Gandalf, by having Frodo "awake early this morning" and then falling asleep again, because the moment was too private and the emotion too deep to give it onlookers. I still think that reunion in Ithilien would have been more emotional than the one we see in the movie, but the movie gives us the "excuse" of the other members of the Fellowship immediately coming in, so Frodo's attention and joy has more than one focus.
A kind of unusual situation here, in that I think we see a microexpression in a real smile by Gandalf (especially in frame 6), but don't really see any from Frodo. But, to be honest, I don't think Frodo's completely at ease. Even in the last frame on this page, his eyebrows are still showing some concern. We get laughter but not a full smile like the one we see from Gandalf. It's interesting that the one elwoodian element that seems to show up in all of Elijah's characters - the tongue forcing itself up behind the front teeth - shows up twice in this shot. We usually see it when the character is afraid or horrified beyond normal emotions. A case could be made here for it expressing a joy that's beyond normal emotions, but if you add in the eyebrows I don't think Frodo's quite sure yet what's happening. That process may also be interrupted by the other members of the Fellowship entering. [Frodo still trying to "work things out" at this point is an important piece of evidence for my theory of what's happening in this scene, as we'll see as we go along.]