Finding the Longest Wavelength for Interference Patterns

[mrjeffy321]

I have a problem in which there are two radio broadcasting towers places a given distance apart (136 m). Each tower is broadcasting at the same frequency as the other one, but the frequency (or rather the wavelength is what I am concerned about) can be adjusted to alter the interference at a point (Q) that is another given distance (41 m) away from one of the towers, making it 177 m distance away from the other tower since it is arranged in a line.

I am looking for the longest wavelength that will cause destructive interference at point Q.
Also, I am looking for the longest wavelength for which there will be constructive interference at point Q.

I know constructive interference occurs when the waves are in sync with each other and thus add, creating a stronger (higher amplitude) wave.
Destructive interference occurs when the waves are 1/2 wavelength out of sync with each other and the waves cancel each other out.

I have a formula to describe the electric field of the waves,
E_1 = A*cos(omega*t + phi)
E_2 = A*cos(omega*t)
where omega is the angular frequency, t is time, and phi is the phase of the wave you begin watching it at.
One can modify the formula using the relationship between omega*t to be,
E = A*cos((2*pi / lamba)x + phi)
where lamba is the wavelength and x is the distance traveled.

So now I am loooking for the [longest] wavelength that can be emmited from tower A, travel 177 m, and either be in sync, or 1/2 cycle out of sync from a wave of the save wavelngth emmited from tower B and only traveling 41 m.

I can't seem to grasp the concept of how to do this and where real numbers fall in.

 

[Claude Bile]

You need to calculate the relative phase of each wave at the point of observation. This will depend on the various distances you mentioned.

Claude.

 

[mrjeffy321]

That still isn't much help to me.
How would I do that? and what then?

 

[quantumdude]

Since both towers and the observation point are all collinear, you can easily figure out the path length difference for the towers. That term should ring a bell, but if it doesn't then you should look it up in your book. That section of your book should also explain what the conditions are on the path length difference so that you observe constructive and destructive interference.

 

[mrjeffy321]

It just occurred to me the correct answer (136 m for constructive, 272 m for desctructive), and it turns out I am right.
But I still can't show it mathmatically if I had to with all that cosine stuff.
I figured that the smallest wavelength to cause interference would either have to be .5 or 1 wavelengths out of phase of the other wave, not 1.5 or 2, or 2.5 or 3, ... since that would mean the wave length is getting short.
So after I knew that for destructive interference, the wave was.5 wavelengths behind, and I knew that it was traveling an extra 136 meters, ta da, the wave length must be 272. and then the same process for constructive interference.


Funny, while I was typing this last reply, Tom Mattson came along and pointed out the very section in the book that gave me this revelation.
R_2 - R_1 = m*lamba

 

[quantumdude]

But I still can't show it mathmatically if I had to with all that cosine stuff.

You could do it using the following trig identity.

$$\sin(\alpha)+\sin(\beta)=2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)$$