Finding the Longest Wavelength for Interference Patterns

In summary, Claude is looking for the longest wavelength that will cause destructive interference at point Q. He knows the wavelength for destructive interference is .5 wavelengths behind and that it is traveling an extra 136 meters. He uses the trig identity to find the wavelength for destructive interference is 272 meters.
  • #1
mrjeffy321
Science Advisor
877
1
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.
 
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  • #2
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.
 
  • #3
That still isn't much help to me.
How would I do that? and what then?
 
  • #4
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.
 
  • #5
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
 
Last edited:
  • #6
mrjeffy321 said:
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.

[tex]\sin(\alpha)+\sin(\beta)=2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)[/tex]
 

1. What are interference patterns?

Interference patterns are a phenomenon that occurs when two or more waves interact with each other. This can happen when waves from different sources overlap, causing areas of constructive interference where the waves reinforce each other, and areas of destructive interference where the waves cancel each other out.

2. What causes interference patterns?

Interference patterns are caused by the principle of superposition, which states that when two or more waves meet, the resulting displacement is the sum of the individual displacements. This results in areas of constructive and destructive interference depending on the relative phases of the waves.

3. What types of waves can produce interference patterns?

Interference patterns can be produced by any type of wave, including electromagnetic waves (such as light and radio waves) and mechanical waves (such as sound waves). However, for interference patterns to be observed, the waves must have a constant frequency and wavelength.

4. How are interference patterns used in science and technology?

Interference patterns have many practical applications in science and technology. They are used in interferometers to make precise measurements, in diffraction gratings to split light into its component wavelengths, and in holography to create 3D images. They are also used in everyday devices such as anti-glare coatings on glasses and noise-cancelling headphones.

5. Can interference patterns be observed in nature?

Yes, interference patterns can be observed in nature. Examples include the colorful patterns on soap bubbles, the iridescent colors of peacock feathers, and the shimmering colors of oil slicks on water. These patterns occur due to the interference of light waves reflecting off of thin layers of different materials.

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