As depicted by this GIF, a dolphin moves its tail in a fluid up and down motion while it is swimming, however, the center of it's body and it's head move very little only slightly tilting up and down with each tail movement.

This video shows a mechanism imitating a birds flapping wings. Although birds and dolphins are hardly similar, the movement of a bird's wings and a swimming dolphin's body are very much alike. This video shows how the amplitude of the tail, or in this case the wing, is increased the further away from the body the tail is, giving this movement a fluid appearance.