How Close Can You Be to Speaker B for Destructive Interference?

[jaded18]

[SOLVED] Interference of Sound Waves

Two loudspeakers, A and B, are driven by the same amplifier and emit sinusoidal waves in phase. The frequency of the waves emitted by each speaker is 172 Hz. You are 8.00 m from speaker A. Take the speed of sound in air to be 344 m/s.
What is the closest you can be to speaker B and be at a point of destructive interference?
__________


I am having the hardest time trying to visualize the problem. I know that destructive interference occurs when the difference in path lengths traveled by sound waves is a half integer number of wavelengths. So I need to know the wavelength of the sound which is just 2m.

I also know that in general if d_a and d_b are paths traveled by two waves of equal frequency that are originally emitted in phase, the condition for destructive interference is d_a-d_b=n(wavelength)/2 where wavelength is what I calculated it to be (2m) and n=any nonzero odd integer. I think I need to know what the value of n is that corresponds to the shortest distance d_b to solve my prob. (is d_a=8m? then what is d_b?)

I'm going around in circles and getting nowhere. Please help!

 

[Astronuc]

d_a=8m is given.

One has determined the wavelength of the sound, 2 m, so the distance 8 m is 4 wavelengths. The the distance to B must be out of phase by 180 degrees in order to destructively interfere.

A half wavelength is 1 m.

 

[Blueshift5]

Hello,

I understand your confusion and I will try my best to help you solve this problem. Let's start by defining some variables:

d_a = distance from speaker A to the point of destructive interference
d_b = distance from speaker B to the point of destructive interference
f = frequency of the sound waves (172 Hz)
v = speed of sound in air (344 m/s)
λ = wavelength of the sound waves (2m)

We know that for destructive interference to occur, the path lengths traveled by the two sound waves must differ by a half integer number of wavelengths. In other words, d_a - d_b = n(λ/2), where n is any nonzero odd integer.

Since we are trying to find the closest distance to speaker B that will result in destructive interference, we can assume that d_a is fixed at 8m (since you are 8m away from speaker A). This gives us the equation:

8 - d_b = n(λ/2)

Now, we need to find the value of n that will give us the shortest distance d_b. Since n is an odd integer, let's start with n = 1:

8 - d_b = (1)(2)

d_b = 6m

This means that if you are 6m away from speaker B, you will experience destructive interference. However, this is not the closest distance possible. Let's try n = 3:

8 - d_b = (3)(2)

d_b = 2m

This is the closest distance to speaker B that will result in destructive interference. Any closer and the path lengths will not differ by a half integer number of wavelengths.

So to answer your question, the closest you can be to speaker B and experience destructive interference is 2m.

I hope this explanation helps you better understand the concept of interference of sound waves. Let me know if you have any further questions.