# Two loudspeakers are 1 meter apart. A person stands 12 meters directly in front of one of the speakers.?

a) Calculate the lowest frequency which destructive interference can be heard at this point.

b) Calculate the lowest frequency which constructive interference will be heard at this point.

Relevance

Constructive interference of two waves at some point in space happens when the two waves are "in sync" at that point.

So put another way, the peaks have to align with peaks and the troughs have to align with troughs.

That means the paths of one wave and the paths of the other wave have to have lengths that differ by a WHOLE number of wavelengths.

Think about it. Say you have two speakers 5 m apart (one behind the other), and the wavelengths of sound they produce are both 5 m.

Then when the wave from the behind speaker reaches the front speaker, it has just traveled exactly one wavelength so it is again matching up in shape with the wave currently going out of front speaker.

Of course we have to assume the speakers are in sync themselves, i.e. the waves are coherent when they get out of the speakers.

Destructive interference happens when the waves cancel out. The troughs have to match with peaks and vice versa.

This happens if the path lengths have a difference of a half-number of wavelengths, like 0.5λ, 1.5λ, 2.5λ, 3.5λ...

Anything between a whole number and a half-number (like λ/3, 3.14λ, 10.2λ...) will produce some interference but it will neither be completely constructive or completely destructive.

So geometry.

The person is 12 m from one speaker and √(12^2 + 1^2) = 12.0415 m from the other (if we assume it's a right angle - the question doesn't say).

The farther speaker produces waves that need to travel 0.0415 m more than the ones from closer speaker.

b) For CONSTRUCTIVE interference to happen, this path difference must consume a whole number of wavelengths.

0.0415 m = k λ (where k is any whole number)

But wavelength is ratio of speed of sound and frequency, λ = c/f

0.0415 m = k * c/f

f = k*c / 0.0415 m

We see smallest frequency occurs when k=1 (i.e. the path difference has to hold exactly one wavelength)

f = 1 * c / 0.0415 m = (340 m/s)/(0.0415 m) = 8.2 kHz

a) DESTUCTIVE interference happens when k = 0.5, 1.5, 2.5, 3.5...

The lowest frequency is again using above formula when k=0.5 (the path difference must hold half a wavelength)

f = 0.5 * (340 m/s)/(0.0415 m) = 4.1 kHz

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