Popular rumor is gain doubles each time elements are doubled. In other words, adding a second vertical above another vertical would add 3dB gain. Adding two more after that (total of four) produces 6dB gain over a single element. The same concept is often applied to stacked Yagi or dipole antennas. In neither case is this true.

Gain actually comes from forcing nulls into areas that used to contain high levels of radiation. The newly created null will remove energy from the null areas, and that energy moves to enhance gain in other directions. This effect is sometimes called pattern multiplication.  (Before antenna modeling software was commonly available, we often used pattern multiplication to estimate patterns and gain. It still is a useful tool.)

The graphs below, from Jasik's Antenna Engineering Handbook, shows the gain of various couplets or elements when placed either end-to-end  (Collinear) or parallel above each other (stacked broadside).

Collinear Gain

First we have the end-to-end or collinear element placement gain.

We can see the gain for two elements peaks at .9 wavelength spacing. This spacing is the current maximum spacing of the elements, NOT the end spacing. With a 1/2 wl dipole in each element, the end-to-end spacing would be .9 - .25 -.25 = .4 wavelengths. The overall array length would be .9 + .25 + .25 = 1.4wl

With two dipoles end-to-end the center-to-center spacing of current maximums would be .25+.25=.5wl. The absolute maximum gain would be found on the graph above at the crossing of the vertical .5 relative spacing line and intersection of curve 2 (two elements). The gain would be less than 1.9dB in ANY collinear antenna.

We would achieve 3dB at about .71-.25-.25= .21wl tip-to-tip spacing, or 1.21wl total collinear element array length. 

To double gain again (adding 3dB more) the array would have to be four elements with .75wl center-to-center spacing in elements! The array would be 3*.75=2.25wl (this is number of elements minus 1 multiplied by required spacing) for the total spacing of four elements). The end-elements would extend .25 wl each from the current maximum, so the array length from end-to-end would be 2.25+.25+.25=2.75wl. 

To go from three to six dB requires we change antenna length from 1.21wl to 2.75wl!

We cannot simply double length to double gain! That concept is wrong.

Note we can get more than 6dB gain by using more than .75wl element center-to- element center spacing. We really should have .95wl element center-to element center spacing, making the array N-1*S + El = L where N=number of elements, S=spacing, El = element length.  The result is an array of four elements would have optimum gain of 6.7dB with a length of 3.35wl.

The longer the array is, the wider the individual elements should be spaced for optimum gain.

Broadside

Broadside gain applies to elements that are parallel and one above the other. This is the OPTIMUM or maximum gain, not the actually gain you might have. It is for DIPOLE elements in freespace, Yagis would require wider spacing to produce the same gain! 

Optimum broadside stacking distance increases with more directive elements or cells. This is why a pair of multi-element Yagis stacked requires wider spacing than a pair of dipoles, and why less maximum stacking gain is possible with the Yagi than we might obtain with stacked dipoles. Think of it this way, if the antenna is already narrow there is less unwanted energy available to move to the main lobe.

Here is the optimum gain graph for dipoles in freespace:

You can see maximum gain occurs at .675wl stacking height. The stacking gain is 4.8db, not 3dB as we often see claimed. Again the more elements the narrower the pattern, and the narrower the pattern the wider spacing must be between elements for maximum gain.

I hope these graphs help dispel the myth that doubling element numbers doubles gain! It just doesn't work that way.

Rules of Stacking

1.) Doubling elements or array size does not guarantee double (3 dB more) gain. That's a myth.

2.) The more narrow initial antenna pattern is, the wider stacking distance becomes for maximum gain improvement.

3.) Optimum stacking distance for gain  is NEVER 1/2 wl, it is always wider.

4.) Optimum stacking distance can be very wide for sharp pattern antennas.

5.) Maximum gain occurs only when a null is forced into an area that used to contain high energy levels.

6.) Height above ground affects antenna pattern, and because of that it also affects optimum stacking distance.