For this collection of observations, I wish I could quickly collect some video fragments to illustrate the point instead of just writing about it, or that I could easily animate some examples. Neither of those are practical at the moment so words alone will have to do.
Snakes are the place to start with this pattern. Snakes move their bodies in a wave like motion which propels them forward along the ground. Alligators (or is it crocodiles -- maybe both) have legs which stick out from the sides of their torso, as opposed to the way mammals have legs underneath their torso. You can see a more subdued wave pattern along the spine of alligators when they move. I think the same is mostly true of how small lizards and geckos move, but they move quickly enough that I can't quite see it with my own eyes. Obviously the legs and feet provide some valuable traction and leverage, but the movement of the torso still exhibits the wave. The same wave is also pretty evident in the way sharks (and other fish) swim. Swimming snakes slither through water much as they do over land -- eels too.
As a kid I was fascinated that dolphins had the same sort of wave, but up and down instead of side to side. I even had a personal kind of classification of the good guys vs bad guys in the animal kingdom based on which way their wave went. Whales and dolphins and sea lions and otters all wave up and down, whereas sharks and eels and barracudas wave side to side. As I try to visualize it now, I'm not entirely convinced that sea lions and otters and walruses wave up and down -- I'll have to look for that next time I'm sitting in front of a nature show on the subject. Thinking of the shape of their hind flippers reinforces the notion, but it's been a long time since I actually looked for it.
Last year while walking our dog, Ellie, I noticed with great surprise that she's got a slight side-to-side wave in her spine when she walks or trots. I can't keep my eyes on her spine long enough to know if it's also the case when she runs, though it seems likely.
A few months ago, Sarah and I rented Winged Migration. I've been meaning to mention it in my blog ever since. I'm interested in the way birds fly and I wish I knew more about migration patterns, so I was naturally fascinated by the film. I was struck by the close-ups of flying geese. Their shoulders move up and down while their heads and tails seem to stay in place. In fact it looks somewhat like watching an accomplished athlete swim the butterfly stroke -- up and down waves.
At park in Toronto, I once noticed that seagulls appear to row through the air. I've tried to see that in other seagulls on other occasions. I can't spot it very consistently. In that park it was quite pronounced.
Segmentation is one of the earliest evolutionary organizations to emerge. This notion is suggested by the many examples spread broadly throughout the animal kingdom. You can see it in earthworms, and in insects. In larger creatures segmentation is most apparent in the spine and rib cage. It's especially apparent in snakes and eels 'cos they're all spine. My casual observations suggest that the more segmentation in the critter, the more apparent the wave in their motion. I think these waves emerge from the segmentation in the anatomy.