Constructive and Destructive Interference

[forestmine]

Homework Statement

Two identical loudspeakers are located at points A & B, 2.00 m apart. The loudspeakers are driven by the same amplifier and produce sound waves with a frequency of 784 Hz. Take the speed of sound in the air to be 344m/s. A small microphone is moved out from point B along a line perpendicular to the line connecting A and B.

(a) At what distances from B will there be constructive interference?
(b) At what distances from B will there be destructive interference?
(c) If the frequency is made low enough, there will be no positions along the line BC at which destructive interference occurs. How low must the frequency be for this to be the case?

Homework Equations

v=fλ
f=nv/2L

The Attempt at a Solution

I'm not sure how to approach this problem. All I know is that constructive interference occurs when the distances traveled by the two waves differ by a whole number of wavelengths, whereas destructive interference occurs when the distances traveled differ by a half-integer number.

Any direction would be greatly appreciated. Thanks!

 

[grzz]

First step: find the wavelength λ.

 

[forestmine]

Ok, that's what I was thinking. And I'd want to use λ=v/f, but I don't have a value for v -- other than the speed of sound through the air but that isn't necessarily v for the sound emitted by the speakers...

 

[grzz]

The speakers are emitting sound. This sound is traveling through the air. The interference is interference between sound waves.
So it is the speed of sound that one requires.

 

[forestmine]

Ok. So, it is appropriate to use 344m/s when talking about speed of sound through the air, but if we're talking about the speed of wave propagation of say a guitar string, it will not necessarily be that same value.

So I calculated .439m for the wavelength...and I'm really not sure where to go from here.

 

[technician]

Do you know the condition for constructive interference in terms of path difference and wavelength.

 

[grzz]

constructive interference: path difference is nλ for integer n and
destructive interference: path difference is (n + 1/2)λ

... as you said.

 

[forestmine]

I think so. For constructive interference the distances traveled by the two waves differ by a whole number of wavelengths. And for destructive interference, they differ by half-integer numbers.

So the wavelength I solved for in this case, is the same wavelength for both speakers A and B? And if that's the case, then for part (a), is it simply 0, 1*.439, 2*.439, 3*.439...etc, or am I missing a step?

Then I would think for destructive interference, it would occur at distance .439/2, [(3)(.429)]/2, [(4)(.439)]/2...etc.

 

[technician]

Can you write down an expression for the path difference from A and B?

 

[technician]

If the distance from B to the first max is x, what is the distance from A to this first max?
remember B is on a line perpendicular to the line AB

 

[forestmine]

Ok, so the distance from B is x. The distance from A would be $\sqrt{x^{2}+2^{2}}$ = the distance from A to x?

 

[technician]

A to x must be 1 wavelength more than x for the first max so
the distance from A to x = x + 0.439

 

[forestmine]

Why is that? I don't understand that

 

[technician]

You are almost there now.
Waves from A meet waves from B to reinforce. The first time this happens is when the path difference is one wavelength so the distance from A to the max must be 1 wavelength greater than the distance from B.
A, B and the max form a right angle triangle so pythagoras relates the distances.
x^2 + 2^2 = (x+0.439)^2... should give you x

 

[technician]

I got x = 4.34m

 

[forestmine]

I got 4.34m for x using the above method you mentioned, and I'm not entirely sure where to go from there...Thoroughly confused about the subject matter, you could say...

 

[Delphi51]

You could do it again with 2λ, then 3λ, ...
Or do it once with path difference nλ and evaluate x for n = 1, 2, 3, ...
And again with (n+.5)λ.

 

[forestmine]

I'm sorry, I don't understand. My thought is to simply do (.439)2, (.439)3. I don't see how speaker A comes into play, since they both have the same wavelength.

 

[Delphi51]

Oh, you want the path difference between A to x and B to x.
That's sqrt(x² + 4) and x. Set that difference equal to nλ and solve for x. Evaluate for as many values of n as you need.

 

[forestmine]

Ok, so solving for x I got 4.34m. Then plugging that back into sqrt(x^2+4) I find the distance between A and x, which is 4.78. Finding the difference -- 4.78-4.34=.44m. Is that the value I use for the wavelength? I still don't seem to be getting the correct answers.

 

[Delphi51]

You found the wavelength in post #5, 0.439.
The x = 4.34 is the first answer for part (a). I put the 1 into a spreadsheet cell and my formula for x in another cell x=4.34 for n = 1. Then I replaced the 1 with a 2 to get another x value for constructive interference. I was surprised to find it was smaller than 4.34. The larger n values give constructive points closer in! And there are only a few of them.

 

[technician]

The distance from B to the max = 4.339m (I use 4 sig figs) This is 10 wavelengths
If you calculate the didtance from A to the max you should get 4.778m and this is 11 wavelengths.
The path from A is 1 wavelength longer than the path fromB.
The waves are meeting in phase and reinforce.
Hope this helps you picture what is going on

 

[forestmine]

I'm taking the difference in distance, .44m, and setting that equal to nλ, and for constructive interference, using whole number integers for n. so .44=nλ, and using that equation, I don't even see what variable I'm solving for. No matter how I manipulate the numbers, I'm not getting the correct answers.

 

[Delphi51]

Do you have a right triangle in your diagram with length 2 on side AB, another side x, and the hypotenuse sqrt(x² + 2)? The path difference is nλ = sqrt(x² + 2) - x.
If you expand that out and solve for x you get
x = (4 - n²λ²)/(2nλ). When you put in n = 1, you get x = 4.34. Put in other values of n to get the other answers for part (a).

 

[forestmine]

That's exactly what I did, however, in solving for x, I can't work it out to the same equation you have above. I'm having some algebra issues, apparently...

 

[forestmine]

Nevermind, silly mistake! Thank you so much for the help!

 

[Delphi51]

Most welcome! Don't worry; it happens to all of us.