a person stands 3.50 m from one speaker and 5.20 m from an identical speaker. If there is a constructive interference where n = 1, what is the frequency?

we first calculate the path difference between the two speakers, which is 1.70 meters. Since �=1n=1 (given in the question), the wavelength (�λ) is also 1.70 meters. Assuming the speed of sound is 343 meters per second, we can use the formula Frequency=SpeedWavelengthFrequency=WavelengthSpeed to find the frequency:

Frequency≈343 m/s1.70 m≈201.76 HzFrequency≈1.70m343m/s≈201.76Hz

So, the frequency is approximately 201.76 Hz.

msm555 msm555 · Answer 2 · 2024-01-26T04:01:41+01:00

Answer:

[tex] 201.76 [/tex] Hz

Explanation:

In constructive interference between two identical speakers, the path difference between the two speakers must be an integer multiple of the wavelength for the waves to reinforce each other.

The formula for the path difference [tex] \Delta x [/tex] between two sources for constructive interference is:

[tex] \Delta x = n \lambda [/tex]

where:

[tex] \Delta x [/tex] is the path difference,
[tex] n [/tex] is the order of the interference (in this case, [tex] n = 1 [/tex]),
[tex] \lambda [/tex] is the wavelength.

For two sources, the path difference can be calculated as the difference in the distances from the person to the two speakers.

Given:

Distance to the first speaker, [tex] d_1 = 3.50 [/tex] m
Distance to the second speaker, [tex] d_2 = 5.20 [/tex] m

The path difference is:

[tex] \Delta x = |d_2 - d_1| \\\\ = |5.20 - 3.50| \\\\ = 1.70 \textsf{ m} [/tex]

For constructive interference ([tex] n = 1 [/tex]), the path difference is equal to one wavelength ([tex] \lambda [/tex]).

[tex] \Delta x = \lambda [/tex]

We can rearrange the formula to solve for the wavelength:

[tex] \lambda = \Delta x = 1.70 \textsf{ m} [/tex]

Now, to find the frequency [tex] f [/tex], we use the wave equation:

[tex] v = f \lambda [/tex]

where

[tex] v [/tex] is the speed of sound.

The speed of sound in air at room temperature is approximately [tex] 343 [/tex] m/s.

Substituting the values:

[tex] 343 \textsf{ m/s} = f \times 1.70 \textsf{ m} [/tex]

Solving for [tex] f [/tex]:

[tex] f = \dfrac{343 \textsf{ m/s}}{1.70 \textsf{ m}} \\\\ = 201.7647059 \\\\ \quad \approx 201.76 \textsf{ Hz} [/tex]

Therefore, the frequency for constructive interference with [tex] n = 1 [/tex] is approximately [tex] 201.76 [/tex] Hz.