How do different path differences affect interference in sound waves?

[Taniaz]

Homework Statement

Two loudspeakers, A and B, are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is L = 2.00 m to the right of speaker A. The frequency of the sound waves produced by the loudspeakers is f = 206 Hz. Consider point P between the speakers and along the line connecting them, a distance x to the right of speaker A. Both speakers emit sound waves that travel directly from the speaker to point P. a) For what values of x will destructive interference occur at point P? b) For what values of x will constructive interference occur at point P?

Homework Equations

Path differences for destructive (n + 1/2 λ) and constructive interference.(n λ)

The Attempt at a Solution

For part a) xa-xb = x-(L-x) = 2x-L = (2l -1) λ/ 2, where l is an integer and then they got x = L/2 + (2l - 1)λ / 4
I don't understand how they got (2l -1) λ / 2 and x = L/2 + (2l - 1)λ / 4
For b) xa-xb= 2x-L = lλ so x = L/2 + l/2 λ

For a I would have simply just done it as 2x-L = λ/2 so x = L/2 + λ/4

 

[Taniaz]

Ok is the destructive interference equation n+1/2 λ or n - 1/2 λ. If it's the latter then the question makes sense but then are both their and my answers correct?

 

[CWatters]

Tip: Draw a sine wave. Then redraw it twice below, once shifted +1/2λ and once -1/2λ.

 

[Taniaz]

They both interfere destructively so it would be +/- 1/2 λ?

 

[Taniaz]

Some people use n+1/2λ and in places they use n-1/2 λ?

 

[Taniaz]

When solving this question: Two identical loudspeakers are located at points A and 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 air to be A small microphone is moved out from point B along a line perpendicular to the line connecting A and B (line BC in Fig. P16.70). (a) At what distances from B will there be destructive interference? (b) At what distances from B will there be constructive 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?

I solved part a using just nλ/2 and not n+1/2 λ or n-1/2 λ

Not sure what to do with part c as to how low the frequency must be.

 

[CWatters]

The lower the frequency the longer the wavelength. For destructive interference the path difference must be at least one half wavelength.

 

[Taniaz]

Thank you.

Some people use nλ/2 or
n+1/2λ and in other places they use n-1/2 λ? What's the difference?