We're not sure that the male swan bows to the female. Actually, the behavior you are seeing would be a dipping of the heads into the water prior to mating and this occurs by both swans as it signals to each swan the acceptance and readiness for mating.

You may also see the swans form a "heart" with their heads/necks before and after the mating. The male swan will then raise up on the water flapping his wings and making vocalizations indicating that the mating was a success. This is called the "Triumphant" sign. The male swans will also exhibit this same "Triumphant" sign wing flapping and vocalization after chasing a possible threat from its habitat. The Regal Swan