A pair of speakers separated by distance d = 0.630 m are driven by the same oscillator at a frequency of 678 Hz. An observer originally positioned at one of the speakers begins to walk along a line perpendicular to the line joining the speakers as in the figure below. (Use v = 343 m/s.)Two speakers are side by side, with one speaker on the left and one on the right. The speakers are separated by a distance d and emit sound waves in the same direction. A man stands directly in front of the speaker on the right but a distance x away from the speaker.(a) How far must the observer walk before reaching a relative maximum in intensity? m(b) How far will the observer be from the speaker when the first relative minimum is detected in the intensity? m
Question
A pair of speakers separated by distance d = 0.630 m are driven by the same oscillator at a frequency of 678 Hz. An observer originally positioned at one of the speakers begins to walk along a line perpendicular to the line joining the speakers as in the figure below. (Use v = 343 m/s.)Two speakers are side by side, with one speaker on the left and one on the right. The speakers are separated by a distance d and emit sound waves in the same direction. A man stands directly in front of the speaker on the right but a distance x away from the speaker.(a) How far must the observer walk before reaching a relative maximum in intensity? m(b) How far will the observer be from the speaker when the first relative minimum is detected in the intensity? m
Solution
(a) The observer will reach a relative maximum in intensity when the path difference is equal to a whole number of wavelengths. The path difference is the difference in distance each wave travels from each speaker to the observer.
The formula for the path difference is: Δx = d*sin(θ), where d is the distance between the speakers and θ is the angle between the observer and the line connecting the speakers.
However, since the observer is moving perpendicular to the line connecting the speakers, θ = 90° and sin(θ) = 1. So, Δx = d.
The wavelength (λ) can be calculated using the formula: λ = v/f, where v is the speed of sound and f is the frequency.
Substituting the given values, λ = 343 m/s / 678 Hz = 0.506 m.
Setting the path difference equal to the wavelength, d = λ, we find that the observer must walk 0.506 m to reach a relative maximum in intensity.
(b) The observer will detect the first relative minimum in intensity when the path difference is equal to a half wavelength.
Using the same formula for the path difference, Δx = d, and setting this equal to half the wavelength, d = λ/2, we find that the observer must walk 0.506 m / 2 = 0.253 m to detect the first relative minimum in intensity.
So, the observer will be 0.253 m away from the speaker when the first relative minimum is detected in the intensity.
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