Interference of microwave transmitters of different wavelengths

[DomPhillips]

This question is about the interference pattern observed when the waves from two microwave transmitters interfere.

The first parts of the question involve the wavelengths of microwaves being the same for each transmitter. For the last part the question proposes that the wavelength of one of the microwave transmitters is halved (from 30 mm to 15 mm). The question asks what observational effect this will have on the interference pattern.

Now at first glance I notice that since the wavelengths are not the same for both microwaves the two microwave sources have different frequencies and are therefore no longer coherent sources, and thus I would intuitively expect no interference pattern to occur. Am I correct? Or is there some new interference pattern that would form in this situation?

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

Welcome to PF;
There will, technically, be an interference pattern - it will look different though.
There is a clue here that the pattern will change with time. The question is: how?

The trick is to imagine you have a detector in a position where the first experiment gets a maximum, then work out what it would see if the wavelength of one source were halved. Repeat for when the detector is at a minima.

 

[DomPhillips]

I'm struggling to see how the interference pattern would change with time since both microwaves are obviously propagating at the same velocity.

Also I used wolfram alpha to try and visualize the resultant wave at different points on the 'interference pattern' I found that if I add two sine waves as follows (where a could take any value from 0 to 1)...

f(x) = sin(x) + sin(2x + a$\pi$)

...then, regardless of the phase difference between the waves, the resultant wave always seemed to have the same overall intensity - i.e. no points of maximum/minimum intensity seemed to form, and certainly no completely destructive interference as is the case with the sum sin(x) + sin(x + $\pi$).

 

[Simon Bridge]

The wavefronts from the two sources are traveling waves.
If the wavelength is different, they do not maintain the same phase relationship - at some times they will be in phase and at others out of phase.

But if you are happy with your analysis - then that is the answer you should put.

 

[HalfLostAndSearching]

I can confirm that your intuition is correct. When two sources of waves have different wavelengths, they are considered to be incoherent and cannot form an interference pattern. This is because interference patterns arise from the superposition of coherent waves, meaning that they have the same frequency, amplitude, and phase.

In the scenario described, the two microwave transmitters have different wavelengths and therefore different frequencies. This means that they are not in phase with each other and cannot produce a stable interference pattern. Additionally, the halving of the wavelength in one of the transmitters further disrupts any potential interference pattern, as the two waves are now even more out of phase.

It is important to note that even though an interference pattern may not be observed, the waves from the two transmitters can still interact and cause other phenomena, such as diffraction or scattering. However, these effects will not produce a distinct interference pattern.

In conclusion, changing the wavelength of one of the microwave transmitters will disrupt the coherence between the two sources and prevent the formation of an interference pattern.