condition of avoiding grating lobe in 1D finite array by null shifting

[yefj]

Hello, In FINITE arrays we need to choose a distance between elements so we will not have grating lobes in the visible range.
there is a method of pushing the null of a grating lobe outside of the visible range by choosing the proper distance shown in the photo bellow,thus pushing the grating lobe itself outside of a visible range.
but the problem is with the main lobe,if we push the grating lobe outside the visible range why does the main lobe stays at its place?
If someone heard of this method i'll be happy to have some mathematical logical explanation of this process.
(not just brute force choosing weights so we get the needing results)
I am looking a solution which uses the distance between elements only.
Thanks.

 

[Geralt_of_Rivia]

Hello
For calculating peak angles in a broadside array (0 phase difference) we have:

So you can shift the grating lobes (m=1,2,...), but the main lobe (m=0) angle never changes in this case.
(Balanis, Antenna Theory(4th ed), p 300)

 

[yefj]

Hello,this formula is good for infinite array.
I am looking for a formula which has N elemrnts.