Typical sidemount FM broadcast transmit antennas such as made in the US by ERI, Dielectric, Jampro and others implement beam tilt by a phase difference in the power applied to the upper vs the lower half of the array.  The formulas usually are considered proprietary by the manufacturer.

The practical effect of using beam tilt in typical FM broadcast transmit antennas may be different than assumed.  Paper #9 at http://rfry.org has more information on this topic.

The delta length of cable between each antenna is a simple formula;
[(Distance between antennas) * sin(beam tilt angle)] / 1.45.
Where
beam tilt angle is the angel from straight out (broadside or boresight).
The 1.45 item is the square root of Teflon dielectric used in most cable. (Teflon dielectric is about 2.1).

There is no frequency in the requirement, this applies for all frequencies, which is really nice.

Here's an example: Four antennas spaced 5 feet apart which makes a 15 foot wide antenna, scanned to angle of 30 degrees off broadside. Use a 4 way power divider and have the length of each successive antenna be different by;
5 feet * sin(30) / 1.45= 5 * 0.5 / 1.45=1.72 feet.

Placing the power divider in the middle of the array of 4 antennas, and you picked a 10 foot cable to the first antenna, the second antenna would have 11.72 feet of cable, the third antenna has 13.44 feet of cable and the 4th antenna has 15.16 feet of cable.
The antenna beam points to the side with the longest cable.

With antenna spacing of half wavelength, you can make your beam scan to very wide angles.

Caution: If you increase your spacing of the antennas beyond one half wavelength (0.5*11.8/fghz inches)and try to scan to wide angles, a second beam pops up in the opposite direction and you have two beams. The inadvertent one is called a "grating lobe". This calculation is all just physical lengths. So if you draw a picture of all the antennas sending energy to 30 degrees off broadside, you simply make the electrical length of the cable (not the physical length) plus airwave lengths add up to be equal for all four antennas in the 30 degree direction. It's just that easy.
kch

RE:  "Here's an example: Four antennas spaced 5 feet apart which makes a 15 foot wide antenna, scanned to angle of 30 degrees off broadside."

COMMENT I believe the original post questions the configuration for beam tilt in the elevation plane. Therefore wouldn't the the array need to be stacked vertically -- not horizontally?

If the beam of a horizontal array of FM antennas needed to be moved horizontally some number of degrees, it is simpler just to re-aim the array.

RE: "Caution: If you increase your spacing of the antennas beyond one half wavelength (0.5*11.8/fghz inches)and try to scan to wide angles, a second beam pops up in the opposite direction and you have two beams. The inadvertent one is called a 'grating lobe'."

COMMENT:  The example of a 4-bay array with 1/2-wave vertical spacing already has a grating lobe above and below the horizontal plane.

If each element in this array is driven with equal power and phase, and has a cos¹ elevation pattern itself, then there will be a sidelobe in the elevation pattern of the array at + and -41° wrt to the horizontal plane, of about 0.19X relative field each.  There will also be elevation pattern nulls at ±30° and ±90°.

It might be noted that the antenna described in the above paragraph has an elevation plane main lobe 3dB beamwidth of about 26°, so it is unlikely that adding elevation plane beam tilt to it would produce much benefit in most installations.

RF

Visit http://rfry.org for FM broadcast RF system papers.

I wrote: "then there will be a sidelobe in the elevation pattern of the array at + and -41° wrt to the horizontal plane,..."

CORRECTION:  The axis angle for maximum relative field in the sidelobes of the example is ±44°.  My apologies...

R. Fry

Grating lobe versus side lobe;  

These are two different items.
Sidelobes are present in all arrays but grating lobes hint a math error.

Sidelobes are usually at least 13 dB weaker than the main lobe.

Grating lobes could be called very high sidelobes beams that are nearly the same amplitude level as the primary beam that you are trying to generate.

If your array has grating lobes, you've spaced the antennas too far apart and need to narrow the spacing. The grating lobe reduces your antenna gain by 2 or 3 dB, plus if you're a radar antenna, it can confuse you as to where the target is.

Grating lobes grate on ones nerves and may hint you've had either a math snafu or have told the technician to space antennas in cm. and he spaced them in inches when he tested them.

kch

"Sidelobes are present in all arrays..."  (Higgler)
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Clarification:  Assuming the same element patterns in each case, no sidelobes exist for 2-bay, 1/2-wave spaced arrays, and arrays of an even number of 1/2-wave spaced bays driven with equal RF phase and binomial power division.

Something is rotten in Denmark; maybe scottyUK’s dead left over fish. Pardon my French. But I have seen dielectric loaded guassian beam horns and hamming weighed –46dB arrays. But all had sidelobes however so modest.  If U have a zero side lobe weighting function, patent it Quick! The signal processing boys will kill for such an anti-aliasing feature.

GOTWW:  "If U have a zero side lobe weighting function, patent it Quick!"
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It's too late. This antenna type has been in use in the broadcast industry for years already. John Kraus describes the general design in Chapter 5 of "Antennas For All Applications," 3rd edition.

A discussion of this antenna configuration, and a NEC-2 surface pattern example are included beginning with slide 20 of paper #10 at http://rfry.org.

I guess that I better get out of Denmark. I am stinking the place up; and the natives are getting restless! Better fly by way of Norway.