The relation that allows us to set the aperture or to choose the distance to take a photo with correct brightness while using a manual flash is this one:
Guide number = f-number × distance, For a certain ISO value.
My question is: why the shutter speed (i.e. time) is not present? It should have influence on that, since longer time = higher brightness and shorter time = lower brightness.
The simple equation assumes that the flash is effectively the only source of light in your image. This is a reasonable assumption in many cases, because since exposure works on an exponential scale, the amount of light from typical indoor room lighting is a drop in a bucket compared to that provided by the flash. See Do flash guide numbers assume some amount of ambient light built-in? and the chart there for a little deeper exploration of this.
That means that the shutter time doesn't matter at all — the exposure is linked to the light output of the flash, and motion is frozen by its (usually very short) duration.
Of course, if you use a longer shutter speed, the relative amount of ambient light will increase and may become a factor. It will also weigh in significantly if you're some distance away or it's relatively bright in the scene (like, you're outside in the day). If you want that not to happen, see What does it mean to "kill the ambient"?.
See How can I take 'balance flash' pictures with studio lights indoors? or What is lighting ratio? for how to calculate the "fill ratio" — the ratio between ambient light and your flash.
Flash duration is typically much shorter than most cameras' flash sync speed.
If the flash only has a duration of, say, 1/1000 second (or 1 millisecond), it matters not if the shutter is open 1/250 second (4 milliseconds) or 1/25 second (40 milliseconds), the energy from the flash that is captured by the photo will be the same in either case.
What shutter duration does affect is the amount of ambient light recorded. But your formula is strictly for calculating the light energy from a flash and does not include anything about ambient light or the ratio between ambient light and flash.
Both of the other answers are correct, but I think there might be an easier way to understand it:
Consider being in a completely black room with no light... the shutter can be open indefinitely because there is no light available. The only thing that matters is that the shutter is fully open when the flash fires so that all of the sensor can see it. The shutter is fully open at any shutter speed equal to x-sync and slower.
The flash power setting provides the higher/lower brightness, distance determines how spread out (strong) it is when it reaches the subject, and the aperture setting regulates how much gets through.
We use the Guide Number as an aid to gauge the f-number setting.
Accept, as an example, your ISO setting is 200. You consult a table of guide numbers and discover it’s 180. Your principal subject is 10 feet downstream of the flash. What would be the appropriate f-number setting? To compute, we divide guide number by distance -- thus 150 ÷ 10 = 15.5. Now round to the nearest full f-stop = f/16.
In modern times, the shutter speed setting is moot because the source of the flash is an electrical discharge of high voltage applied to a glass tube filled with xenon gas. A super short blitz of light ensues. The blitz is so quick that few, if any shutters have any ability to mitigate. Thus the real problem is synchronization. Now we are talking the timing of the applied charge so that the shutter will be fully open when the blitz occurs. There are exceptions when using specialty high-speed cameras.
The published guide number is just a good guestimate. Several difficult to predict factors come into play. To name a few: Indoor or outdoor scene – height and color of indoor ceiling – reflectivity and contrast of scene - size and shape to flash reflector – plus many more.
To refine the guide number for an actual shoot: On scene, shoot a series at different f-stops. The idea is to discover a setup that delivers a faithful image. Pretend f/11 works. Now multiply the principle subject distance by f-stop used. Say 10 feet @ f/11 – revised guide number is 10 x 11 = 110. Setting for synchro-sunlight: For flash fill in sunlight, likely best is flash subordinate to sunlight by 1 f-stop. We adjust flash-to-subject distance to achieve. Better method is to adjust flash power if possible. If not, set shutter speed to synchronize. Set camera f-number for sunlight exposure. Divide guide number by principal subject distance. Pretend f/11 with guide number of 120.
Now divide 120 by 11 = 11 (rounded ). This is distance in feet flash to principal subject. At this distance, flash intensity mirrors sunlight exposure. Now multiply this distance by 1.4 = 16 (rounded). This is flash to subject distance, one f-stop subordinate.
I’m happy to have found this thread. The number of YouTube videos providing false information of incident light meters work in situations that are 100% flash lit, is disturbing. Typically they’ll set their light meters to 1/250 shutter speed (magic number?) while also rightfully stating (sometimes) that shutter speed only needs to slower than sync speed and doesn’t affect lighting.
One way to think about this is as forth. In an externally lit scene (say natural light) where you set your camera to 1/250th of a second and choose say an ISO of 100 (assuming that’s what you are shooting), then an incident light meter in front of the subject can give you the f-stop needed for the 18% gray reading. The simplicity of this scenario is that the light energy is practically flat during the exposure. Basically a rectangle function over 4 ms. And your shutter speed determines the width of this window.
However when you use a flash, the profile of the light intensity is not flat. It’s more like a Gaussian function with the bulb heating up to slowly emanate light and then reach peak light and then dim down. The entire width of this energy burst may only be a fraction of a millisecond.
However the light meter is basically integrating all this energy. And if it is a matter of “tricks of the trade” to input the shutter value as 1/250 in the meter (regardless of the fact that you could set the shutter to be 1/10 or 1/30 or 1/60, I.e. long enough to guarantee flash event happens in that window) it can only mean one thing. The area under the intensity curve of this particular manual flash burst should equal the area of a rectangle of width 4ms (magic number window) times the intensity one would expect in a natural light scene where the intensity didn’t fluctuate.
TL;DR
While guide numbers are useful for distance based determination of appropriate f-stops, perhaps flash manufacturers can provide an equivalent-flash shutter speed that we as practitioners could blindly input as shutter speed while metering purely flash lit scenes.
