Wave Race is all about racing Kawasaki branded jetskis through a variety of waterways, which you could probably have gleaned from the title. But, nestled within the cartridge is one ride that I'm pretty sure Kawasaki doesn't yet have a patent on.
Okay, so go to Stunt Mode and pick Dolphin Park.
Then complete the course by doing all the stunts (handstand, spin across the handlebars, stand up (and do backflip), a left barrel roll, a right barrel roll, a backflip, and an underwater dive), while simultaneously passing through all the rings on the course. A pretty tall order, actually.
If you don't know how to do some of these moves, I'm pretty sure they're detailed in your manual, which you totally still have, right?
But if you're successful, you'll hear dolphin noises, and you'll notice that the riders are riding dolphins on the title screen, a good sign.
So now pick Championship mode, normal difficulty, and warm-up. While you're choosing your driver, hold Down (the direction) on the control stick (or pad, it doesn't matter which).
Once you get to Dolphin Park, you should be riding on your very own dolphin.
You can do pretty much anything on your dolphin that you could do on the Jetskis, barrel rolls and back flips are pretty fun.
There are a couple of tricks you can't do, like standing up on it, but I think that's understandable.