Waves question.... constructive inteference

[toesockshoe]

Homework Statement

Two radio antennas transmit identical signals in phase and in all directions of frequency 120 MHz . The distance between the antennas is 8.95 m . Let the origin be at one antenna, and let x be a distance from the origin toward the other antenna.

Homework Equations

Look at picture

The Attempt at a Solution

Work done in attachment.
I'm how to solve the problem. To get the maximum constructive interference, m and n would need to be whole number right? There are no whole number solutions to the equation I set up though.

 

[Simon Bridge]

You have worked out the position of the wave crests at a particular time ... but the waves travel: over time the crests will travel past/through each other - so, at the time you picked, perhaps full constructive interference is just not happening.

The condition you want is that the path difference is a whole number of wavelengths.
This will give you only one whole number to work with.

 

[toesockshoe]

You have worked out the position of the wave crests at a particular time ... but the waves travel: over time the crests will travel past/through each other - so, at the time you picked, perhaps full constructive interference is just not happening.

The condition you want is that the path difference is a whole number of wavelengths.
This will give you only one whole number to work with.

the problem says the two signals are introduced in phase. shouldn't the path difference be 0 everywhere?

 

[Merlin3189]

Can I ask just what the question is?
It seems to me that you are only looking along the line of the two antennae and they will never be in phase along this direction.

 

[toesockshoe]

Can I ask just what the question is?
It seems to me that you are only looking along the line of the two antennae and they will never be in phase along this direction.

Oh my apologies. I forgot to state the question.
Question:
Find the values of x at which constructive interference between the transmitted waves would be found.
Enter your answers in ascending order separated by a comma.

Also... why would they never be in phase? The problem says they start out in phase... so arent they always in phase?

 

[Merlin3189]

I'm not sure what your equation represents, since I'm not sure what m and n are. Perhaps you could draw a diagram and mark on the point which is 2.5 x m from one antenna and 2.5 x n from the other.

 

[toesockshoe]

I'm not sure what your equation represents, since I'm not sure what m and n are. Perhaps you could draw a diagram and mark on the point which is 2.5 x m from one antenna and 2.5 x n from the other.

m and n are whole numbers. The only point of maximum constructive interference are when both signals are integer multiples of their wavelengths away from their source (their antennas). The m and n represent any whole numbers.

 

[Merlin3189]

#5 came while I was writing.
They are in phase at the antennae. But when the signal from one antenna reaches the other, it will be delayed by the time it takes to travel 8.95m and will now be out of phase. As that wave and the one from second antenna are now travel together, they can never get back into phase in that direction.
Which now shows me what the question is driving at! My original comment is in fact wrong. It is only true along that line once you get past the space between the antennae.

Your comment about m and n now makes sense to me. They do not have to be whole numbers. (You can see that easily by looking at the point where the signals must obviously be in phase.)

 

[Merlin3189]

The only point of maximum constructive interference are when both signals are integer multiples of their wavelengths away from their source (their antennas).

No this is not so. If they start in phase (as they do here) they will be in phase anywhere where the path difference is a whole number of wavelengths plus the same fraction of a wavelength. So for eg they would be in phase if one had gone 3.5 waves and the other had gone 4.5 waves.

 

[toesockshoe]

No this is not so. If they start in phase (as they do here) they will be in phase anywhere where the path difference is a whole number of wavelengths plus the same fraction of a wavelength. So for eg they would be in phase if one had gone 3.5 waves and the other had gone 4.5 waves.

I said MAXIMUM constructive interference. Isn't this what the question means?

 

[toesockshoe]

#5 came while I was writing.
They are in phase at the antennae. But when the signal from one antenna reaches the other, it will be delayed by the time it takes to travel 9.5m and will now be out of phase. As that wave and the one from second antenna are now travel together, they can never get back into phase in that direction.
Which now shows me what the question is driving at! My original comment is in fact wrong. It is only true along that line once you get past the space between the antennae.

Your comment about m and n now makes sense to me. They do not have to be whole numbers. (You can see that easily by looking at the point where the signals must obviously be in phase.)

Well in any way, how can I solve for m an n? My equation has an infinite number of solutions if they don't need to be whole numbers. Is my equation set up wrong?

 

[Merlin3189]

Essentially you can ignore whole numbers of waves, because the wave is identical to itself when shifted by any whole number of wavelengths.
Max constructive interference then occurs when they have both traveled the same extra distance.

Have you found one easy point where the two waves will be in phase yet?

 

[toesockshoe]

Essentially you can ignore whole numbers of waves, because the wave is identical to itself when shifted by any whole number of wavelengths.
Max constructive interference then occurs when they have both traveled the same extra distance.

Have you found one easy point where the two waves will be in phase yet?

can i set m to 2 and n to 1.58? This would give me an x of (2.5 * 2) or 5. But this can be done with any arbitrary point. I could have also set m to 1.99, and it would have given me another x value.

 

[Merlin3189]

If you use the equation 2.5m = 8.95 - 2.5n then m an n are not whole numbers, but they must have the same fractional part.

So, eg if the antennae were 7.5 apart and λ= 3, you could have, say m= 0.25 and n= 2.25,
so that: dist from ant A = x= 0.25 x 3 = 0.75 and dist from ant B = (7.5-x) = 2.25 x 3 = 6.75

In fact I found it much easier (on this simple example) to just draw the diagram and mark in the points where the waves must be in phase.
I did make up an equation, so that I could be in tune with your approach, but I did not find that as easy. I stayed with m and n as whole numbers, but added a fraction for the final bit.

I suspect PF experts might prefer to use a cos function, but that does not make it easier for me.

 

[Merlin3189]

Just noticed I'm talking as if the antennae are 9.5 metres apart, but it should be 8.95 metres. So I've gone back and edited them.

 

[toesockshoe]

Just noticed I'm talking as if the antennae are 9.5 metres apart, but it should be 8.95 metres. So I've gone back and edited them.

i still don't understand... why doesn't my equation have an infinite number of solutions? How do I figure out which m and n values I should use?

 

[Merlin3189]

can i set m to 2 and n to 1.58? This would give me an x of (2.5 * 2) or 5. But this can be done with any arbitrary point. I could have also set m to 1.99, and it would have given me another x value.

But if m=2 and n= 1.58, then the fractional part is different. You would need 2.58 and 1.58 (which is not a solution) or 1.99 and 2.99 (which is also not a solution)
You were correct to say that the waves would be in phase if m and n were whole numbers, but that gave you no solutions on the line.
So then we have to look for solutions where m and n are NOT whole numbers. So then there is another condition. You can't just use any numbers for m and n.
The waves are in phase after they have gone any whole number of λ, but they are also in phase after they have BOTH gone 0.5λ, or both gone 0.7λ, or both gone 0.3421λ, etc. Also they are in phase if one has gone 0.3λ and the other 1.3λ or 2.3λ or 3.3λ etc. So the new condition is that the fractional part is the same.

So if you call the fractional part k, then the two distances are n+k and m+k where m and n are again whole numbers.

 

[Simon Bridge]

The usual approach is to formulate the problem in terms of the path difference.
You get constructive interference from two in-phase sources where the path difference is a whole number of wavelengths.
This is where you have been getting confused.

You've probably seen the derivation done for Young's interference ... check that.

So why not work out the equation for how far point x is from each source, then subtract one from the other?