What is the Halachic or Rabbinical source for the difference in washing hands for cleanliness (right-left alternating three times) and for washing for a meal (right-right, left-left)? Why is there a difference?

Washing one's hands in the morning requires pouring water on each of the hands three times. It is not sufficient to pour a triple quantity on each hand at one time. Theoretically, if one were to pour water three times on his right hand, stopping between each pouring, and then three times on the left hand, stopping between each pouring- the hands would be considered purified.The necessity of pouring water three times on each hand stems from the purpose of the washing- to overcome the spirit of impurity that attaches itself to a person asleep at night (Shulchan Aruch Orach Chayyim 4 2; Mishna Berura 4 10; Beur Halacha 4). Washing one's hands before meals is required for a different purpose and pouring water on one's hands once is sufficient. Pouring the water a second time is just to remove the first water and if a large quantity of water was used the first time the second pouring is still recommended by some but not required (Shulchan Aruch Orach Chayyim 262 2 Mishna Berura 262 17-21).