As you can see from the photo, President Obama's bow to the Emperor Akihito was deeper than that of Japanese Prime Minister Hatoyama.
Of course, in Japan the more subservient the person, the more deeply she (it tends to be a she) must bow to the more powerful person. As this video about Japanese bowing indicates, Obama did the "waiter bow".
Obama's bow ought to earn him points for cultural sensitivity. But he loses points for having overdone the gesture. Moreover, perhaps Obama should lose points for having done anything more than nod his head to the emperor. I had thought the protocol of Americans bowing to royalty ended with a certain revolution.
On the bright side, at least he didn't throw up.
UPDATE 1: Two further reflections.
First, concerning etiquette, from having lived in Japan, I came to understand that it is not considered good form to combine a bow with a handshake, as Obama is doing. The emperor graciously extended his hand, a bow was not called for. Moreover, after I came to appreciate the range of meanings attached to a bow, I found it preferable always simply to shake hands. The bottom line for me was that a Japanese bow carried implications that are not easily reconcilable with egalitarian values -- which brings me to my next point.
Second, I would argue that America's great challenge is not to exhibit humility (as many claim), but to remain true to its founding principles. America was founded on the premise that all human beings are created equal. The notion of a hereditary monarchy stands in conflict with this principle. An American president honors his country's values and sets an example for the world by bowing to no man. Such symbolism is important to many people. For example, even today, a Thai can be sent to jail for years simply for not standing before the image of a hereditary head of state. Traveling abroad, the president of the world's most powerful republic should stand tall for the alternative moral order his country represents. Even those who don't happen to support certain American values -- i.e. Japanese monarchists -- can respect a country that is true to its principles. A lot of anti-Americanism these days probably has far less to do with any perceived lack of humility than profound disenchantment with a government that fails to live up to the expectations the country sets for itself.
UPDATE 2: "It's just a sign of respect. Therefore, Obama's sloppy bow was of no consequence."
Does the Japanese bow "simply" convey respect for the other as an individual? I beg to differ. Few forms of greeting are less simple than a Japanese bow.
A Japanese bow reflects social status. A bow is one of various of daily life performances -- polite language (keigo) being another -- by which Japanese pay respect to their respective relational identities within the social hierarchy. The enactment of these rituals throughout daily life reinforces the hierarchy. It's an exhibition of power relations.
Obama's bow was both incorrect and inappropriate in view of his social stature. As head of state, Pres. Obama is Emperor Akihito's equal, but his bow did not reflect their equal status. (As leader of the world's most powerful country, Obama could just as easily be considered emperor's superior from the super-hierarchical Japanese perspective.)
The bottom line is that the "forms" of social interaction tend to matter to Japanese people -- that's why they go to so much trouble with them! They should also matter to Obama's advisers and US embassies, whether in Tokyo, Beijing, or Kabul. That's respect. It's not being culturally-sensitive to prove your world-leading country can't summon a competent coach for its president. Either take the trouble to get it right, or don't bother.
Better a handshake next time.