The 24^th State blog does a review of bowing between leaders of countries, putting President Obama's deep bow to the Saudi King in context. (hat tip: Clarice Feldman) Particularly relevant are pictures of bows between other national leaders at international gatherings, demonstrating that nearly 90 degree bows with bent knee, as practiced by Obama before the Saudi monarch (but not the British one) are virtually unknown. 

Notice he does a full bow, with his right leg going back to support his weight as his knee bends.  Bending your knee is a supremely different bow then bending from the waist.  It's a practiced bow you give to a man you consider your superior.  It's not a sign of respect. It's a sign of subservience.

Go ahead and try it yourself.  Bow from the waist.  Then bow as our President did.  Not so easy, is it?

... there was nothing wrong with Mr Obama's display of deference.

"I think it was a sign of respect and in no way diminishes the ranking of the President or indicates the greater strength of the one being bowed to," Ms Starr said.

"I bowed, wore the attire, ate the figs as a sign of respect when I was in Saudi.

"I applaud the President for showing this courtesy."

... the "thou need not bow" commandment from the State Department's protocol office maintained a constancy of more than 200 years. Administration officials scurried to insist that the eager-to-please President had not really done the unthinkable.

"It was not a bow-bow, if you know what I mean," said Ambassador Molly Raiser, the chief of protocol.

White House officials described Mr. Clinton's tilt as something of an improvisation. Because Emperor Akihito broke with tradition in turn to raise his glass at the state dinner, some even said Mr. Clinton had managed something of a breakthrough.

"Presidents don't bow, and Emperors don't toast," one official said. "So this was a little bit like the cultures meeting each other halfway."