# Destructive and Constructive Interference for Sound Speakers

## Homework Statement

http://i.imgur.com/FiYb9OE.png
Problem b). Looking at the attached document of my teacher, the visual representation of the answer does make sense.

## Homework Equations

2pi(delta(x1-x2))+delta(phase constant)
Basic interference problem in One-Dimension.

## The Attempt at a Solution

After all, the two waves end up contructively interfering. However, in the picture, the phase constant of 2 is pi/2 and the phase constant of 1 is 0. My teacher uses the equation 2pi(delta(x1-x2))+delta(phase constant). However, here is what I do not understand. For delta(phase constant), she substracts as follows: phase constant of 2 (pi/2) minus phase constant of 1 (0). Inevitably, she gets a positive value for delta(phase constant).

Why is it that she can change the order of the delta? "1"-"2" vs "2"-"1"
I looked at the proof of this formula to dig up more info. For those interested, I looked up chapter 21.6 of Knight's Physics textbook. The author clearly mentions that when one chooses a particular order, he must stick to it for both delta x and delta phase constant.

I tried sticking to it. But the result gives me destructive interference!
Why?

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Can no one answer this? It's a pretty basic question.

Doc Al
Mentor
Why is it that she can change the order of the delta? "1"-"2" vs "2"-"1"
The ##k\Delta x## is really ##kx_2 - kx_1##, which is the additional distance that wave 2 must travel. Perhaps your instructor is using ##x_1## and ##x_2## to represent the location of the speakers. That would explain why she would use ##\Delta x = x_1 - x_2##.

Make sense?

Doc Al, not really.
I got this answer from someone else:

"The problem is in how you have calculated the $$\Delta \phi_o$$ term. As you have pointed out, if you have used $$\Delta x=x_1-x_2$$ then $$\Delta \phi_o=(\phi_o)_1-(\phi_o)_2$$ and, in fact, this is true.

If you examine the picture in part (b) and say that wave #1 has a phase of zero, then wave #2 has a phase of $$-{\pi \over 2}$$. If it had a phase of $$+{\pi \over 2}$$ as you have assumed, then wave #2 would look like the graph of $$-sin(\theta)$$ when in fact it looks like $$+sin(\theta)$$.

Therefore you have:
$$\Delta \phi_o=0-\big(-{\pi \over 2}\big)={\pi \over 2}$$
Using this, the equation:
$$\Delta \phi=k(x_1-x_2)+\Delta \phi_o$$
gives constructive interference."

What does he mean? y=sin(x-pi/2) at x=0 has a trough, NOT a crest as shown on the picture for speaker 2.

Doc Al
Mentor
As you have pointed out, if you have used $$\Delta x=x_1-x_2$$ then $$\Delta \phi_o=(\phi_o)_1-(\phi_o)_2$$
No, it depends on what you mean by ##x_1## and ##x_2##. What do you think they mean?

Well, they meant the position of x1 and x2. I don't understand your question.

Doc Al
Mentor
Well, they meant the position of x1 and x2.
The position of what?

I think when you are clear on the meaning of x1 and x2, then the problem will be clear.

The position of the speakers no?

Doc Al
Mentor
The position of the speakers no?
Yes and no. That's what makes this so confusing--the notation is misleading.

In the equation shown by Knight, he compares the phase of one wave (##kx_1 + \phi_1##) with that of a second wave (##kx_2 + \phi_2##). In these expressions, ##x_1## and ##x_2## do not represent the positions of the speakers; they represent the distance from the speaker to some common point of overlap where you want to find the phase difference. So ##x_1## is the distance from speaker 1 to that point; similarly, ##x_2## is the distance from speaker 2 to that point. Thus if speaker 2 is further away, ##\Delta x = x_2 - x_1## will be positive.

But now express that difference in terms of the positions of the speakers along the x-axis, where ##x_1## and ##x_2## now stand for the positions of the speakers, and you'll get ##\Delta x = x_1 - x_2## because being further away means a smaller value of x. You didn't magically change the order of subtraction, you just changed the meaning of the symbols.