Interference broadcast transmitter help

[ScienceMonkey]

Need some help getting started...

Two radio transmitters are placed 50 m apart on either side of a runway. The two transmitters will broadcast the same frequency, but out of phase with each other. This will cause a nodal line to extend straight off the end of the runway. As long as the airplane's receiver is silent, the pilot knows she's directly in line with the runway. If she drifts to one side or the other, the radio will pick up a signal and sound a warning beep. To have sufficient accuracy, the first intensity maxima needs to be 60 m on either side of the nodal line at a distance of 3.0 km. What frequency should you specify for the transmitters?

I know that there will need to be completely destructive waves. And for that, ΔL/λ=0.5, 1.5, 2.5... But in order for her to be directly in line with the runway, ΔL=0, so this would not be true. Also, I think this, ΔL/λ=0.5, 1.5, 2.5, is only true when Φ=0, 2π, 4π, etc., but Φ will need to be equal to π for being completely out of phase. I can't see how knowing this helps because ΔL to be directly in line with the runway would still be equal to zero, or is my thinking completely flawed?

 

[nrqed]

Need some help getting started...

Two radio transmitters are placed 50 m apart on either side of a runway. The two transmitters will broadcast the same frequency, but out of phase with each other. This will cause a nodal line to extend straight off the end of the runway. As long as the airplane's receiver is silent, the pilot knows she's directly in line with the runway. If she drifts to one side or the other, the radio will pick up a signal and sound a warning beep. To have sufficient accuracy, the first intensity maxima needs to be 60 m on either side of the nodal line at a distance of 3.0 km. What frequency should you specify for the transmitters?

I know that there will need to be completely destructive waves. And for that, ΔL/λ=0.5, 1.5, 2.5... But in order for her to be directly in line with the runway, ΔL=0, so this would not be true. Also, I think this, ΔL/λ=0.5, 1.5, 2.5, is only true when Φ=0, 2π, 4π, etc.,
:

That`s right. But if you give a difference of phase of Pi between the emitters, then the bisecting line going through the point exactly between them will be a line of destructive interference...a minimu. which is what you want here. In a case like this, the usual formla for ΔL/λ are switched. You will have *maxima* for ΔL/λ=0.5, 1.5, 2.5... (because the difference of path will cancel the difference if phase of the emitter, giving constructive interference.)

Hope this helps

Pat

but Φ will need to be equal to π for being completely out of phase. I can't see how knowing this helps because ΔL to be directly in line with the runway would still be equal to zero, or is my thinking completely flawed?

 

[ScienceMonkey]

So that means that what was normally constructive, (ΔL/λ=0,1,2...) will now be completely destructive since the transmitters are out of phase? So I could still use ΔL/λ=0,1,2... at the bisection?

 

[lightgrav]

it's a bisector ... the ΔL = 0 ! What you want is the first CONSTRUCTIVE angle, right next to the bisector.

 

[nrqed]

So that means that what was normally constructive, (ΔL/λ=0,1,2...) will now be completely destructive since the transmitters are out of phase? So I could still use ΔL/λ=0,1,2... at the bisection?

At the bisection, ΔL is still equal to 0, obviously. The point is that the condition for constructive vs destructive is switched compared to the case of emitters in phase. So the points where ΔL/λ=0,1,2... correspond to destructive interference now (as opposed to the more usual case).