Destructive and Constructive Interference for Sound Speakers

[alingy1]

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?

 

[alingy1]

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

 

[Doc Al]

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?

 

[alingy1]

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]

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?

 

[alingy1]

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

 

[Doc Al]

Well, they meant the position of x1 and x2.

The position of what?

I don't understand your question.

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

 

[alingy1]

The position of the speakers no?

 

[Doc Al]

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.