I'll be glad to give it a try in his absence... :-).
If you're interested in the gorey details, read the blurb at the end.
Otherwise here is some actual data (as given in Pietropaolo's book):
Three cases: (all 4-foot fluorescent bulbs)
A) 4 40W std cool-white bulbs, 2" from white reflector, 2" centers, new
B) 4 40W std cool-white bulbs, 2" from white reflector, 2" centers, 200 hrs.
C) 2 40W std cool-white bulbs, 2" from white reflector, 2" centers
Distance from foot-candles for
tube (inches) case A case B case C
------ ---- ---- ----
1 1800 1600 1100
2 1600 1400 860
3 1400 1300 680
4 1300 1100 570
5 1150 940 500
6 1000 820 420
7 900 720 360
8 830 660 330
9 780 600 300
10 720 560 280
11 660 510 260
12 600 480 240
18 420 320 130
24 260 190 100
> Three feet will result in a considerable reduction in light intensity.
You get about an 8:1 reduction in light going from 1" to 2 feet.
This graph is particularly nasty when you consider that full sunlight
is about 10,000 foot candles. You can only get about 1/5 of full
sunlight intensity even at one inch from the tube. At one foot distance
you are down to 1/20 of full sunlight intensity, even with a 4-tube fixture.
Notice that the light doesn't fall off as fast as the 1/r^2 that you'd
expect from a point source. If you had an infinite line emitter, you would
expect it fall off as 1/r, and an infinite plane source would not fall off
at all with distance.
You can see this in the data above. Going from 1" to 2" causes very
little fall off of intensity because the source seems like an infinite
plane source. A bit further distance causes a linear fall off, and when
you are further away than the width of the array, you tend towards
an inverse square fall off.
The bottom line is that an array of many tubes (quasi-plane emitter) falls
off more gradually than a single tube (line source) or (horrors!) a
single incandescent bulb (point source).
> One way I've gotten around this is to prop up smaller plants by
> placing their pots on some sort of support. The pots may be lowered
> as the plant grows, always keeping the apex only a short distance
> from the light.
This works well only for plants that don't have a large vertical
dimension. Otherwise it is too dim at the base of the plant as
compared to the growing tip.
Another way to greatly improve the drop-off rate is to put reflectors
around the sides of your growing tank. In the limit of not having any
absorbing plants, the light distribution becomes uniform.
-- Rick----------- Gorey Details -------------
Or: "How to calculate intensity vs distance for rectangular fluorescent light arrays"
You can treat a fluorescent light as a lambertian source, which means that the intensity at a point away from the fixture is the 2d integral of light over the surface of the light fixture.
The light from any given point varies with the square of distance from the bulb and as the cosine of the emission angle.
So the intensity at a point on the z axis, being illuminated by a rectangular lambertian patch of light in the x,y plane with x1 < x < x2, and y1 < y < y2 is given by the integral:
integral of y from y1 to y2 { integral of x from x1 to x2 { cos (theta) / (x^2 + y^2 + z^2) } dx } dy
cos (theta) is equal to (z / sqrt (x^2 + y^2 +z^2)), so combining the two equations gives the integral:
integral of y from y1 to y2 { integral of x from x1 to x2 { z / (x^2 + y^2 + z^2)^(3/2) } dx } dy
Which can be solved to give the intensity versus distance (z) as a function of the light emitter dimensions in x, and y:
B(z) = 4 * A * arctangent( x2*y2 / (z * sqrt(x2^2 + y2^2 +z2^2)))
Which assumes that x1 == -x2 and y1 == -y2, and A is a constant that accounts for the brightness of the light source.
For common fluorescent tubes, a value of A between 200 and 300 gives a good fit to experimental data, and provides a value B(z) in units of foot-candles.