Dance Timing and Rink Acoustics: A Thought Experiment
An interesting question is how ice rink and P.A. system acoustics affects perceived dance timing at the judge's stand.
In particular, the time-of-arrival of the music at the dancer's position in various parts of the rink and the time-of-arrival at the judge's stand could differ by as much as .2 seconds.
(That is an extreme case, in which the speakers are all located at the judge's stand, or at the opposite corner.)
This means that you can be perfectly in sync to the music that you hear, but will appear to be ahead of or behind the music at the judge's stand, depending on your location in the rink.
(It also means that as you traverse the pattern, a metronome you carry won't sync with the music you hear.)
If you wanted to produce a perfect result at the judge's stand, you would have to measure the difference between the time-of-arrival of sound at various parts of the rink surface, to the time-of-arrival at the judge's stand, and compensate the timing of your dance. For a competition with a large audience, the measurements would be somewhat different when the audience is in the stands.
I bet no one does this. It's hard enough to be in sync with your partner!
So this is only a thought experiment, not a practical suggestion...
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I don't know much about acoustics, but here are some things that may or may not be completely correct:
In a modern concert hall, echoes, and the muddy sound they produce, as well as the sound suppression at specific locations (by destructive wave interference) they produce, may be suppressed by a combination of architectural shape, sound absorbing materials in or on the floor, walls and ceiling, draperies and curtains, and absorption of sound by bodies and clothing.
It is expected that a well designed P.A. system will attempt to minimize differences in time-of-arrival of sounds from the original source, the various speakers, and unsuppressed echoes, throughout the area of the audience. The P.A. system may also attempt to suppress residual echoes. An ideal (but impossible) hall in which sound arrives simultaneously at all points is "acoustically dead".
It is more complicated for a physically large music performance group, like an orchestra or large band, because they are large enough for sound to make syncing the group difficult.
It isn't always considered desirable to suppress all echoes in a concert hall: some "reverberation" is musically desirable. That is, a concert hall should be somewhat "acoustically live". The amount desired depends on the tempo and style of the music - e.g., clear voice requires greater clarity, and less reverberation.
Acoustics depends on the audience size and state of dress. So a good sound person must be employed to adjust the P.A. system at each event, and check for proper sound in various parts of the audience. That's why you hear "sound checks" before a concert.
Movie theaters do this too, but it is possible (this may be completely wrong!) they also try to sync the on-screen timing with the time-of-arrival of the sound. This goal is inconsistent with the goal of eliminating muddy sound throughout all parts of the audience, so a compromise may be reached.
For these and other reasons, the P.A. system of a good concert hall or movie theater is much more complex than a typical home hi-fi system, and requires substantial expertise to operate well.
I think that most ice rink buildings, and the P.A. systems they employ, largely ignore these considerations. Some rinks do better than others, but none I've been in do a good job at music and voice clarity, let alone simultaneity.