Question:

Two speakers are placed 2 meters apart and emit sound waves with a frequency of 500 Hz. The speed of sound in air is 343 meters per second.

a) Determine the distance between two consecutive points of constructive interference created by the sound waves emitted by the two speakers.

b) If a person is located 4 meters away from one of the speakers, will they experience a standing wave? Justify your answer.

Answer:

a) The distance between two consecutive points of constructive interference, known as the constructive interference distance (λc), can be calculated using the formula:

λc = λ / (2n)

Where:

• λ is the wavelength of the sound wave
• n is the order of the constructive interference (n = 1, 2, 3, ...)

To find λ, we can use the formula:

λ = v / f

Where:

• v is the speed of sound in air
• f is the frequency of the sound wave

Substituting the given values:

v = 343 m/s f = 500 Hz

Therefore, λ = (343 m/s) / (500 Hz) = 0.686 m (rounded to three decimal places).

Now, let's calculate the distance between two consecutive points of constructive interference for n = 1:

λc = (0.686 m) / (2 * 1) = 0.343 m

Therefore, the distance between two consecutive points of constructive interference is 0.343 meters.

b) To determine if a person located 4 meters away from one of the speakers will experience a standing wave, we need to calculate the distance of the person from the nearest node of the standing wave.

The distance from the speaker to the person is 4 meters, which we'll call d.

Based on the formula for the distance from a node in a standing wave, we have:

d = n * λ / 2

Where:

• n is the order of the node (n = 0, 1, 2, ...)
• λ is the wavelength of the standing wave

Since the person is located 4 meters away from the speaker, we have d = 4 m.

Let's calculate the wavelength of the standing wave:

λ = v / f

Substituting the given values:

v = 343 m/s f = 500 Hz

Therefore, λ = (343 m/s) / (500 Hz) = 0.686 m (rounded to three decimal places).

Now, let's substitute the known values into the formula for the distance from a node:

4 m = n * (0.686 m) / 2

Simplifying:

8 m = n * 0.686 m

Solving for n:

n = 8 m / (0.686 m) ≈ 11.65

Since n should be a whole number, we can conclude that n = 12.

Therefore, the person will experience a standing wave because they are located at the 12th node of the standing wave.

Note: The exact position of the nodes and antinodes in a standing wave is more complex, but this calculation assumes a simplified scenario where the distance between the speakers is much larger than the wavelength of the sound waves emitted.