Interference of Multiple Waves: How to Determine Phase for Maximum Intensity?

[emdezla]

Homework Statement

Four identical wave sources (s1,s2, s3,s4 ) are located at the corners of a square.
We know the phase at three sources: 0 at s1, π/4 at s2, π/2 at s3. Whet is the phase we have to give to s4 in order to have a maximun of intensity at the center of the square?

Homework Equations

δ = k*Δr - Φo

δ -> phase angle
k -> wavenumber
Δr -> distance difference
Φo -> phase at the source

The Attempt at a Solution

[/B]
The main problem I have is that I don´t know how to apply the formula when we have more than two sources. It's sure that Δr = 0 because all sources are at the same distance to the middle, but I don't know how to express Φo

I also know that the intensity is maximised when δ = 2mπ

Thank you very much for your help!

 

[kuruman]

... I don´t know how to apply the formula ...

You don't need a formula. Just think. (a) What must be the phase difference for constructive interference? What about destructive interference? (b) Look at the phases of the waves that are given to you. Do you see any that interfere constructively or destructively?

 

[berkeman]

Don't we need the length of the sides of the square in terms of wavelength, or is that not needed?

 

[kuruman]

Don't we need the length of the sides of the square in terms of wavelength, or is that not needed?

I don't think it's needed. If one thinks "phase difference", the answer will be the same even if the side of the square is less than one wavelength.

 

[berkeman]

Thanks @kuruman

 

[haruspex]

I also know that the intensity is maximised when δ = 2mπ

Are you saying that is the given answer to this problem? Doesn't look right to me. Or is that for two sources?

 

[emdezla]

You don't need a formula. Just think. (a) What must be the phase difference for constructive interference? What about destructive interference? (b) Look at the phases of the waves that are given to you. Do you see any that interfere constructively or destructively?

Okey, I'll try to think.

For constructive interference, the phase difference has to be 2mπ whereas for destructive interference, the phase difference has to be (2m+1)π.

However, I can't see any wave in the problem interfering constructively or destructively.

 

[emdezla]

Are you saying that is the given answer to this problem? Doesn't look right to me. Or is that for two sources?

That has to be the total phase diifference for any given number of sources in order for the intensity to be maximised.

 

[haruspex]

That has to be the total phase diifference for any given number of sources in order for the intensity to be maximised.

How are you defining total phase difference for, say, three sources?
If free to set the phases, the intensity would be maximised when there are no differences. (A phase difference of 2π is the same as no difference. It doesn't make sense to discuss differences of 2π or greater.)

 

[emdezla]

How are you defining total phase difference for, say, three sources?

That's my main problem lol. I don't know how to express the total phase difference for more than two sources.

 

[haruspex]

That's my main problem lol. I don't know how to express the total phase difference for more than two sources.

No, your problem is trying to apply a principle that has no meaning with more than two sources, all but one of which has a given phase.

There are two ways to attack this problem, depending on how long you are expected to take.

The quick way is to use common sense and a sketch. Adding up the three given waves, where does it peak? What additional wave correlates most closely with it?

The other is a standard maximisation process using calculus. It's not hard if you exercise some judgment in discarding irrelevant terms. The general result is quite pretty.
First step, write an expression for the resulting effective amplitude. Hint: it is an integral.

 

[BvU]

Laziness sparks creativity (translation of an old dutch saying ---- so I must be very creative ) The advice by @kuruman is a lot more attractive to me than embarking on an optimization problem a la @haruspex .

As @berkeman discovered after a little nuddge, the size of the square doesn't matter. Easy to see if you imagine zooming in or out (with the same wavelength).

So, whatever you find, it should hold all the way to a zero size square. Kuru carefully hints at that, but I'm a bit more direct.

See that and Robert is your uncle.

 

[haruspex]

The advice by @kuruman is a lot more attractive to me than embarking on an optimization problem a la @haruspex .

I do not see them as alternatives.
The simplification that you and kuruman imply has to be made first. Having done that, the two options I lay out in post #11 apply. My second option has the advantage of addressing the general case, and as I posted, the result is quite pretty.

 

[kuruman]

My second option has the advantage of addressing the general case, and as I posted, the result is quite pretty.

I didn't think of addressing the general case until you pointed it out. It is pretty indeed. Thanks.