Wave Confluence: Ship & Radiotowers (Check Needed)

[Const@ntine]

Homework Statement

A ship floats across the coat, at a distance d = 600 m from it. The radio of the ship receives simultaneously signals of the same frequency from RadioTowers A & B, which are L = 800 m apart. At Point G (Γ), the two waves confluent in a strengthening way, where G's (Γ) distances from A & B are the same (r_A = r_B). At point D (Δ), which is right across B, the signal reaches its First Minimum. Find the wavelength λ.

Homework Equations

Δr = n*λ , n = 0,1,2,3... (Strengthening-A'=2)
Δr = n*(λ/2), n = 1,3,5... (Weakening-A'=0)

The Attempt at a Solution

When the book says "First Minimum arrives at..." in every case it means Δr = n*λ/2 with n = 1. So i went ahead and found rA & rB for Δ through the forming triangles, put them in the above formula and got λ = 140 m. Which is wrong. I kinda figured that since I didn't use the stuff for Γ, but anyway. I turned back and saw that the answer was λ = 800 m. Which is a pretty big number.

So I thoughout of it this way: The signals arrive at Γ, the composite wave starts the way a y = Asin(...) wave would (no Initial Phase so it goes from "zero" to +A and then back down) and at Δ it's where the composite wave reaches the "horizontal axis" again, ie it covers λ/2 in length. Through that and the info about Γ (used the rs to find ΑΓ = ΓΒ = 400 m) I did λ/2 = 400 m <=> λ = 800 m

Now the result is correct, but I have no idea if the whole thing has any logic behind it. I was just trying stuff until I got the correct result. I mean, in every exercise thus far "First Minimum" just means Δr = λ/2.

Any help is appreciated!

 

[haruspex]

found rA & rB for Δ through the forming triangles, put them in the above formula, Δr = n*λ/2

It's not clear to me what you did, but it does not sound right. Please post the details.

 

[Const@ntine]

It's not clear to me what you did, but it does not sound right. Please post the details.

Yeah, I checked it again this morning and figured out my mistake. I messed up on the Pythagorean Theorem. But to break this down:

At Δ we have the "First Minimum", so the signals that arrive from A & B there are out of phase, and result in the Amplitude A of the combined wave being 0. Thus, Δr = n*λ/2, in regards to Δ. But it's the 1st Minimum, so n = 1. Now I need to find r_A & r_B, or ΑΔ & ΒΔ. ΒΔ is d = 600 m. For ΑΔ we construct the triangle with ΑΔ as the hypotenouse, as seen here:

So, ΑΔ = sqrt(L² + d²) = sqrt(800² m² + 600² m²) = 1000 m

Now we have Δr = |r_A - r_B| = | 1000 m - 600 m | = 400 m = 1*λ/2 <=> λ = 800 m

And that's it. I messed up the Pythagorean at first and then everything spiralled from there. I still have a question though: Why tell me that in Γ the two waves confluent in a strengthening way/A' = 2A? I didn't use it anywhere. Couldn't it have just given me the info about Δ?

 

[haruspex]

Why tell me that in Γ the two waves confluent in a strengthening way

Because that tells you that the transmissions are in phase. E.g. if there were 180 degrees out of phase then it would have been destructive interference at Γ and reinforcing at Δ.

 

[Const@ntine]

Because that tells you that the transmissions are in phase. E.g. if there were 180 degrees out of phase then it would have been destructive interference at Γ and reinforcing at Δ.

I figured as much, but I wasn't sure if it was hiding something else. Just to recap though: Because they arrive at the same time and I know that ΑΓ = ΓΒ, that means they're in phase. Otherwise, I'd get a different result. And thus when I go to Δ and find that Δr = n*λ/2, I don't have to factor any phase differences. Got it.

I have a question though: Doesn't the Δr = ...λ/... give me my answer already? If for example I was given the distance of Δ from Α & Β, plus λ. But I didn't know the confluence there. Then say I just put Δr & λ in both formulas, and for argument's sake, let's say that there was a reinforced interference at Δ (so Δr = n*λ). Would I be able to deduce whether the waves are out of phase or not?

There is an exercise example in my book, where there are there are two speakers. A receiver stands in such a way that the 1st Minimum arrives there, so Δr = λ/2. Then, in the "what if" section, it says that if the waves had a phase difference of λ/2, and due to them being placed in such a way that Δr, in regards to the receiver's position, equals λ/2. So that means that at the receiver the two waves have areinforced interference.

I get that on a theoretical level, but how would that work on a problem? Which formula would I use? Δr = nλ/2 or Δr = nλ? Technically it's the first one, but according to it there should be a destructive interference, which isn't true. Are there any other formulas to use in circumstances where the waves are not in phase? I seem to remember something like that from High School, but my book doesn't have anything. I get the theory part, but don't know how to "work" this in problems.

 

[haruspex]

A receiver stands in such a way that the 1st Minimum arrives there, so Δr = λ/2.

Only if they were emitted in phase.
In general, 2πΔr/λ tells you the phase shift. This has to be added to whatever the initial phase difference was.

 

[Const@ntine]

Only if they were emitted in phase.
In general, 2πΔr/λ tells you the phase shift. This has to be added to whatever the initial phase difference was.

Ah yeah, now I'm starting to remember. Any link to where I can read about that? Wikipedia is pretty unreliable. It seems my book has only the "in phase" occurances, and I'm afraid I'm not learning all that properly.