Question

Two loudspeakers, Speaker A and Speaker B, are placed 4.0 meters apart. The speakers are emitting sound waves of the same frequency and in-phase. The speed of sound in air is 343 meters per second.

a) Calculate the wavelength of the sound waves emitted by Speaker A and Speaker B.

b) A person is standing 6.0 meters away from Speaker A. Will the person experience constructive or destructive interference at this location? Justify your answer.

c) At a certain point located 2.0 meters away from Speaker A and 2.0 meters away from Speaker B, the person experiences destructive interference. What is the phase difference between the waves emitted by Speaker A and Speaker B at this location?

Answer

a) The formula to calculate the wavelength is given by:

wavelength = velocity / frequency

Given: Velocity of sound in air (v) = 343 m/s Frequency (f) = (same for both Speaker A and B)

Since both speakers are emitting sound waves of the same frequency, they have the same wavelength.

Thus, the wavelength of the sound waves emitted by Speaker A and Speaker B is given by:

wavelength = velocity / frequency = 343 / frequency

b) To determine whether the person will experience constructive or destructive interference at a distance of 6.0 meters from Speaker A, we need to consider the path difference between the waves emitted by Speaker A and Speaker B.

The path difference (∆x) between two waves can be calculated using the formula:

∆x = distance between the sources * sin(θ)

Given: Distance between Speaker A and Speaker B = 4.0 meters Distance of the person from Speaker A = 6.0 meters Speed of sound in air (v) = 343 m/s

Using the given distances, we can find the value of sin(θ):

sin(θ) = opposite/hypotenuse = (6.0-4.0)/6.0 = 2.0/6.0

sin(θ) = 1/3

Now, we can calculate the path difference:

∆x = 4.0 * sin(θ) = 4.0 * 1/3 = 4/3 meters

Since the path difference (∆x) is equal to half a wavelength (λ/2), the person will experience destructive interference. This is because the path difference is equal to an odd number of half wavelengths, which results in the cancellation of waves.

c) Given: Distance between Speaker A and Speaker B = 4.0 meters

At a certain point located 2.0 meters away from Speaker A and 2.0 meters away from Speaker B, the person experiences destructive interference. This means the path difference (∆x) between the waves emitted by Speaker A and Speaker B is equal to half a wavelength (λ/2).

Let's assume the phase of Speaker A is 0° (or 0 radians). The phase difference between the waves emitted by Speaker A and Speaker B at this location is:

Phase Difference (δϕ) = (∆x / wavelength) * 2π = (2.0 / wavelength) * 2π

Since the path difference (∆x) is equal to half a wavelength, we can substitute it:

Phase Difference (δϕ) = (0.5 * wavelength / wavelength) * 2π = π radians

Therefore, the phase difference between the waves emitted by Speaker A and Speaker B at this location is π radians.