Right-Brained Guide to Combining the Guide Number of Multiple Flashes

What's the guide number of two identical flashes?  That's easy enough.  It's about 1.4 times the guide number of one flash.  But what if you need to know the combined guide number of several different flashes with different guide numbers?  That's easy too -- just count the photons.  (I'm just half-kidding.)

If you have two or more speedlights or are thinking of getting a few, and you'd like to know what the combined guide number is supposed to be, I promise that no matter how much you may hate math, you'll be able to calculate it using a regular (not scientific) calculator after the jump.  I'll also promise to use a common sense approach and keep the math minimal.

BACKSTORY
Last week, I posted about studio strobes and although I know that studio strobes are more cost-effective when it comes to sheer power, I still want the versatility of using multiple speedlights (hotshoe flash).  I decided to add a couple of flashes to my gear - an SB-26 and another SB-600, with the thought of getting a studio strobe later if I find it absolutely necessary.  (The SB-600 was hard for me to swallow because I had purchased an extra brand new one a few months ago but returned it unused and now I bought a used one at the same price... argh!)

Guide Number - A Review

The guide number (GN) of a flash is a measure of its light output, especially for speedlights (not so much studio strobes due to other variables).  It's not an arbitrary number.  Rather this meaningful measurement can tell you the distance that the flash can adequately illuminate a subject at a particular aperture.  If at 10 feet, you have enough flash power that you can use a fairly narrow f/11 aperture, then the GN is 10 feet x 11 or 110 feet.  (More precisely, the GN is the f-number of the aperture times the distance.)

To make GNs comparable, it is important to make sure that GNs are based on the same ISO, same zoom setting on the flash, and same unit (such as feet or meters).

Before we get down to the business of combining guide numbers, let's take a quick detour.

Inverse Square Law - A Review for Non-Mathematicians
The inverse square law tells us that the intensity of a light source diminishes rapidly as we move away from the light source.  But have you wondered why the inverse square law applies to light (more precisely, to a point light source)?  It's not because the light disappears (any grade school kid knows that energy cannot be destroyed).  Rather it's because the light energy gets spread out.  You can almost imagine the tiny photons from a glowing sphere of light shooting out into space in straight lines with the straight lines spreading further and further apart from each other as the photons travel farther away from their home.

The photons spread out horizontally, so we divide the number of photons (i.e. the intensity of light) by the distance to account for the horizontal spread.  If they spread out horizontally only then we wouldn't need the inverse square law.  It would just be called the inverse law. :)  But of course the photons will also spread out vertically, so we have to divide the horizontally spread-out photons vertically as well by dividing the intensity with the distance again, to account for the vertical spread.  In other words, we divide the intensity by the distance squared.  Some might even say the intensity is inversely proportional to the square of its distance. :)

What's this gotta do with combining the GN?  Well as I said, to combine GNs we just count the photons.

COMBINING GUIDE NUMBERS
We could re-imagine GN as a measure of the photons a flash can put out.  Let's take the Nikon SB-800 as an example.  Its GN is 125 feet (35mm zoom, 100 ISO).  Wow - 125 feet is pretty far.  What's even more amazing is that for the light to reach 125 feet while taking into account the inverse square law, we would need 125 x 125 or 15,625 "photons" to get there.

Now suppose we had an SB-800 (GN: 125 feet), two SB-600s (GN: 98 feet each), and an SB-26 (GN: 118 feet).  Lots of photons there.  Let's count them:
SB-800: 15,625 photons
SB-26: 13,924 photons
first SB-600: 9,604 photons
second SB-600: 9,604 photons
Total: 48,757 photons.

How far could 48,757 photons go?  Well, thanks to the darned inverse square law, we have to spread those photons vertically and horizontally by getting the square root of 48,757.  Turns out that's around 220 feet.  Well, there sir (or madam) is your combined GN.

To recap:
1. Determine the guide number of each flash, at same zoom, same ISO, same unit. e.g. 35mm ISO 100, in feet.
2. Count the "photons" of each flash (by squaring the distance).
3. Add the "photons" of the flashes.
4. See how far the total photons can reach under the inverse square law by getting the square root.
You're done!

If you want to double-check your calculations, try out this spreadsheet.

EPILOGUE
Going back to why I got two more flashes instead of a studio strobe, how does a GN of 220 feet compare to the power of a studio strobe?  According to the Paul Buff Alien Bees website, their B1600 fitted with a standard 7-inch reflector (80 degree coverage) has a GN of 236 feet.  So, on paper, if we forget coverage for the moment, the combined GN of my 4-flash setup delivers almost as much light intensity as a B1600.

Note though that at 35mm, the speedlight coverage is only 45 degrees vertically and 60 degrees horizontally according to Nikon.  If we use an 11-inch reflector on the B1600 with 50 degree coverage, then the GN jumps to 450 feet.  The 4-flash setup GN of 220 feet would be more like a B400 with an 11-inch reflector.  So I still have an excuse to get a studio strobe in the future. :)

Meanwhile, how would I trigger all 4 speedlights and control them simultaneously?  Triggering options for a multi-flash setup will be an upcoming post.

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