Two identical loudspeakers are 2 m apart. A person stands 4.6 m from one speaker and 3.4 m from the other. What is the lowest frequency at which destructive interference will occur at this point?
To find the lowest frequency at which destructive interference will occur at the given point, we need to first determine the path difference between the two loudspeakers.
The path difference is the difference in the distances traveled by the sound waves from each loudspeaker to the point of interest. For destructive interference to occur, the path difference needs to be an odd multiple of half the wavelength.
Let's calculate the path difference:
1. First, find the total distance between the two loudspeakers: 2 m (since they are 2 m apart).
2. The person is 4.6 m from one speaker and 3.4 m from the other. Therefore, the path difference is the absolute difference between these two distances: |4.6 - 3.4| = 1.2 m.
Now, we need to find the lowest frequency at which this path difference corresponds to destructive interference. To do this, we can use the equation:
Path Difference = (2n + 1) * (λ/2),
where "n" is an integer and "λ" is the wavelength.
For destructive interference, we need the path difference to be an odd multiple of half the wavelength. So, we can rearrange the equation to solve for the lowest wavelength:
λ/2 = Path Difference / (2n + 1).
Since we are interested in the lowest frequency, we can substitute the speed of sound equation:
Speed of Sound = Frequency * Wavelength,
to solve for the lowest wavelength:
Wavelength = Speed of Sound / Frequency.
Substituting this back into the rearranged equation:
(Speed of Sound / Frequency) / 2 = Path Difference / (2n + 1).
Simplifying further, we have:
Frequency = Speed of Sound / (2 * Path Difference * (n + 0.5)).
Now we can plug in the values:
Speed of Sound = 343 m/s (at standard temperature and pressure).
Path Difference = 1.2 m.
n = 0 (for the lowest frequency).
Frequency = 343 / (2 * 1.2 * (0 + 0.5)).
Calculating this, we find:
Frequency = 143.958 Hz.
Therefore, the lowest frequency at which destructive interference will occur at the given point is approximately 143.958 Hz.