Two small speakers are driven by a common oscillator at 7.60 ✕ 102 Hz. The speakers face each other and are separated by d = 1.27 m. Locate the points along a line joining the two speakers where relative minima would be expected. (Use v = 343 m/s. Choose one speaker as a reference and enter your answers as positive distances from this speaker, from smallest to largest. Enter NONE in unused answer boxes.) m m m m m m
Question
Two small speakers are driven by a common oscillator at 7.60 ✕ 102 Hz. The speakers face each other and are separated by d = 1.27 m. Locate the points along a line joining the two speakers where relative minima would be expected. (Use v = 343 m/s. Choose one speaker as a reference and enter your answers as positive distances from this speaker, from smallest to largest. Enter NONE in unused answer boxes.) m m m m m m
Solution
The minima occur at points where the path difference between the two speakers is an odd multiple of half the wavelength.
First, we need to find the wavelength (λ) of the sound. We can use the formula for the speed of sound, v = fλ, where f is the frequency and v is the speed of sound.
Given: f = 7.60 x 10^2 Hz v = 343 m/s
Rearranging the formula to solve for λ gives us λ = v/f.
Substituting the given values, we get λ = 343 m/s / 7.60 x 10^2 Hz = 0.4513 m.
The path difference for minima is given by (n + 1/2)λ, where n is an integer (0, 1, 2, 3, ...).
We need to find the values of n for which the path difference is less than the distance between the speakers (d = 1.27 m).
Setting (n + 1/2)λ < d and solving for n gives us n < (d/λ) - 1/2.
Substituting the given values, we get n < (1.27 m / 0.4513 m) - 1/2 = 1.81.
The only integer values of n that satisfy this inequality are n = 0 and n = 1.
Substituting these values into the formula for the path difference gives us the locations of the minima:
For n = 0, the minimum is at (0 + 1/2)λ = 0.4513 m / 2 = 0.2257 m.
For n = 1, the minimum is at (1 + 1/2)λ = 1.4513 m / 2 = 0.7257 m.
So, the minima are expected at 0.2257 m and 0.7257 m from the reference speaker.
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