Constructive Interference of Sound waves

[amit25]

Homework Statement

A pair of speakers connected to the same signal generator face each other, one at x=0 and the other at x=1.5m. If they are producing a sound frequency of 2000Hz what are the points (position x) of constructive interference between the speakers? Express your answer in terms of λ

Homework Equations

r2-r1=nλ

The Attempt at a Solution

I calculated the wavelength from v=fλ
λ=343m/s / 2000Hz
λ=0.1715m

then from r=L-nλ/2

n=0 r=1.5/2=0.75m

n=1 r=1.5-.1715/2=0.66m

n=2 r=1.5-2(0.1715)/2= 0.58m

 

[TSny]

Not sure if you had a question here. The problem says to find values of x for constructive interference. So, I think it would be best to state your answer as values of "x" rather than "r". (You didn't actually state what "r" represents.)

The question also says to express your answers in terms of λ. [I think you did this, except it would be best to write it as x = (L-nλ)/2]

Did you find all the answers? Do you think negative values of n would yield possible answers?

 

[amit25]

i think r is suppose to represent distance. The question gives me the formula in terms of r
I guess my question is if my calculations are correct? Not sure if I am on the right track. I am not sure how many answers to give I wasnt told but I calculated up to 3.

 

[TSny]

i think r is suppose to represent distance.

What distance? Can you describe what r represents?

The question gives me the formula in terms of r.

I didn't see any formula given in the question.

I guess my question is if my calculations are correct? Not sure if I am on the right track. I am not sure how many answers to give I wasnt told but I calculated up to 3.

Well, I think you are to find all the values of x between the speakers that correspond to constructive interference. Since one speaker is at x = 0 and the other is at x = L, you need to find all constructive interference locations x that lie between 0 and 1.5 m.

It seems to me that your equation x = (L-nλ)/2 is essentially correct . You can write this as x = L/2 - nλ/2. For n = 0, you get x = L/2. Thus, the midpoint between the speakers is a point of constructive interference. This makes sense because the sound travels the same distance from both speakers to get to this point. (The "path difference" = 0).

You can keep plugging in other integer values of n to get more values of x. As you can see, increasing n by 1 moves the point of interference a half wavelength toward the left speaker. If n gets too large, x becomes negative and therefore does not correspond to a point located between the speakers.

You should also be able to see that you can plug in negative values for n. Or, equivalently, you could write x = L/2 +nλ/2 for positive values of n. Now each successive value of n moves you a half wavelength toward the speaker on the right. When n gets too large, x will be larger than 1.5 m and therefore does not represent a point between the speakers.

[The general idea is that if you are located at a point of constructive interference, then you can get another point of constructive interference by moving a half wavelength to the right (or to the left). That's because you will move a half wavelength closer to one speaker while at the same time moving a half wavelength farther from the other speaker. So, you will have changed the "path difference" of the sound by one full wavelength.]

 

[Nickie]

Therefore, the points of constructive interference between the speakers are at x=0.75m, x=0.66m, and x=0.58m. This is because at these points, the difference in distance between the two speakers (r2-r1) is equal to a whole number multiple of the wavelength (nλ), resulting in constructive interference.