A microphone is located on the line connecting two speakers that are 0.546 m apart and oscillating 180degrees out of phase. The microphone is 2.00 m from the midpoint of the two speakers.

What is the lowest frequency that produces an interference maximum at the microphone's location?
What is the next lowest frequency that produces an interference maximum at the microphone's location?

It is not clear where the mic is? a radius of 2 meters is a lot of loci.

The idea here is to find a relationship between the distances to each speaker that results in some integer n + 1/2 wavelengths difference in paths.

To find the lowest frequency that produces an interference maximum at the microphone's location, we can use the formula for constructive interference in a two-source system.

The formula is:

dsinθ = mλ

where:
d = distance between the sources (in this case, the speakers)
θ = angle between the line connecting the speakers and the line from the midpoint of the speakers to the microphone
m = order of the interference maximum (m=0 for the first maximum)
λ = wavelength of the sound wave

First, let's calculate the angle θ:

tanθ = (0.546 m/2) / 2.00 m
θ = tan^(-1)((0.546 m/2) / 2.00 m)

Using a calculator, we find:
θ = 0.266 radians

Now, to find the lowest frequency that produces an interference maximum, we set m = 0:

0.546 m*sin(0.266 radians) = 0 * λ

Simplifying the equation, we get:

λ = (0.546 m*sin(0.266 radians)) / 0

Since m = 0, the wavelength λ can be any value, which means all frequencies will produce constructive interference at the microphone's location. There is no specific lowest frequency.

Similarly, the next lowest frequency that produces an interference maximum can also be any frequency.

To determine the lowest and next lowest frequency that produce an interference maximum at the microphone's location, we need to use the concept of interference in waves.

Interference occurs when two or more waves overlap or superpose with each other. In the case of the question, we have two speakers and a microphone, which means we are dealing with sound waves.

The key to solving this problem is to understand that at the location of the microphone, we want the waves from the two speakers to constructively interfere, resulting in an interference maximum.

The condition for constructive interference is that the path difference between the two waves is equal to an integer multiple of the wavelength. Mathematically, this condition can be expressed as:

Δx = nλ

Where Δx is the path difference, n is an integer, and λ is the wavelength.

In this scenario, the microphone is equidistant from the two speakers, which means the path difference will be zero. Therefore, we can set up the following equation:

0 = nλ

To find the lowest frequency that produces an interference maximum, we need to find the longest possible wavelength that satisfies this equation. The longest possible wavelength is given by:

λ = 2d

Where d is the distance between the two speakers.

Substituting the given values, we have:

λ = 2 * 0.546 m = 1.092 m

Now we can solve for the lowest frequency using the formula:

f = v / λ

Where v is the speed of sound.

Assuming the speed of sound in air is approximately 343 m/s, we can calculate the lowest frequency as:

f = 343 m/s / 1.092 m = 314.51 Hz

Therefore, the lowest frequency that produces an interference maximum at the microphone's location is approximately 314.51 Hz.

To find the next lowest frequency, we can use the same formula:

f = v / λ

However, in this case, we need to find the next longest wavelength that satisfies the equation Δx = nλ. The next longest wavelength is given by:

λ = 2d / (n+1)

Substituting the given values, we have:

λ = 2 * 0.546 m / (0+1) = 1.092 m

Solving for the next lowest frequency:

f = 343 m/s / 1.092 m = 314.51 Hz

Therefore, the next lowest frequency that produces an interference maximum at the microphone's location is also approximately 314.51 Hz.