The deep bow is the highest form of salutation which was more common during the feudal times in Japan. It is unusual to see it used in today's Japan. You will definitely see it when in audience to the Emperor. In the dojo the deep bow should not be used. Oh, the deep bow is referred to as, "Sai-keirei (sigh-kay-ray).
The formal bow is the medium one. How it is done is the arms are extended downward along the legs with the hands resting on the legs above the knees. When you bow you bend at the waist, the head and neck remain in alignment and the eyes remain straight ahead while the body bends about 45-degree angle. It is held for about two or three seconds. [note: no where in any of the descriptions does it indicate that the bow must remain at 45-degree's until the more senior person returns the bow, etc.]
Now, the medium bow (30-degree angle) is the one used when greeting and/or meeting seniors. It is used to show a special kind of respect to the senior or when one is expressing strong feelings such as sorrow, humility or simply apologizing to someone. What is important to remember is that if you encounter that senior several times in the same day, you greet them with the proper medium bow the first time that day and drop back to the light bow thereafter.
The bow most used and I believe most used in the dojo as well is the light bow. The body is bent as described above but at a 15-degree angle instead. It is held for about a second or so and the hands are down at the sides, not above the knees. Even tho the hands are incidental for this bow it is more polite to make the effort to bring them down to your sides. [note: I believe this is more appropriate even in the dojo]
Sometimes you will observer a casual nod of the head in lieu of the light or medium bow. This may be done when in a hotel or restaurant or other places where the staff regularly bow to guests.
Often I have observed over use of the bow in dojo, training halls. I think this is incorrect simply due to inaccurate conveyance as to the use and purpose of the bow. The only variance I understand that should be applied in the training hall is a use of the junior-senior medium bow required at the beginning and end of the training session although decorum in general it is used at the start while the light bow is used for the remainder of the time that day in the dojo.
Other than the generalizations indicated above there is some latitude in bowing. As most things in Japan it is dependent on many cultural requirements that make Japanese - Japanese. With this said, we are not Japanese and we are not in Japan (or Okinawan for that matter).
p.s. one small note, all that I have posted regarding the why we do this in relation to culture, customs and beliefs, these are things that apply most strictly to being in Japan, with Japanese and out of courtesy to Japanese. In the end, if you wish to practice the traditional way of martial arts then a well-meaning and well-informed attempts is ok.
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