We have two waves with the same wavelength that have a path difference. Why does the path length difference have to be an integer multiple of the wavelength in order to obtain constructive interference? It would be enough at least for two plane waves.
Can someone explain both with math and intuitively please?
And why we can have the phenomena of interference just between waves of the same wavelength?
All electromagnetic fields interfere. If two different source produce a field at a position then the fields add up. Note that they add up vectorially. Thus opposite and equal fields cancel: they are out of phase. If the two fields have different frequency then this condition only happens at distinct points in time. For two fields to permanently add up to zero they have to be if the same frequency therefore of the same wavelength.
Two waves with the same wavelength that have a path difference can be represented by two sinusoids with a phase difference between them. The phase difference is a function of the frequency or wavelength and wave propagation speed in addition to the path difference, so you need to specify these along with the amplitude of each wave.
For the equations used to add two sinusoids of the same frequency but with a phase difference look at: https://www.dsprelated.com/showarticle/635.php The derivation of the equations are also given there. As you can see its rather involved, and I do not think there is a simple intuitive explanation for the general case.
Re. "Why does the path length difference have to be an integer multiple of the wavelength in order to obtain constructive interference?"
If the path length difference is an integer multiple of the wavelength the two sinusoids would be in phase (zero phase shift between them) and add together in a trivial manner giving total constructive interference.
For example $Asin(2\pi\omega t) + Bsin(2\pi\omega t) = (A+B)sin(2\pi\omega t)$.
That is because $sin(2\pi\omega t +n\lambda)=sin(2\pi\omega t)$ where $\lambda$ is the wavelength and n is an integer.
