Moe, Larry, and Curly stand in a line with a spacing of d = 1.40m . Larry is 3.00m in front of a pair of stereo speakers 0.800m apart, as shown in the figure(Figure 1) . The speakers produce a single-frequency tone, vibrating in phase with each other.
What are the two lowest frequencies that allow Larry to hear a loud tone while Moe and Curly hear very little?
The two lowest frequencies that allow Larry to hear a loud tone while Moe and Curly hear very little are the frequencies that create a standing wave between the speakers and Larry. The two lowest frequencies are given by the equation f = v/2d, where v is the speed of sound and d is the distance between the speakers. In this case, the two lowest frequencies are f1 = 343 m/s / (2*1.4 m) = 96.07 Hz and f2 = 343 m/s / (2*3.0 m) = 48.03 Hz.
To find the frequencies that allow Larry to hear a loud tone while Moe and Curly hear very little, we need to consider the concept of constructive and destructive interference of sound waves.
Constructive interference occurs when the path difference between two waves is an integer multiple of the wavelength, resulting in reinforcement of the waves and a loud tone.
Destructive interference, on the other hand, occurs when the path difference between two waves is a half-integer multiple of the wavelength, which results in cancellation of the waves and a very little sound heard by Moe and Curly.
Let's start by finding the path difference between the waves reaching Larry from the two speakers.
The total path difference (Δd) between the two speakers and Larry can be calculated using the formula:
Δd = L2 - L1
Where:
L1 = Distance between Larry and the first speaker
L2 = Distance between Larry and the second speaker
Given:
L1 = 3.00 m
L2 = 0.800 m
Δd = L2 - L1
= 0.800 m - 3.00 m
= -2.20 m
The negative sign indicates that the path from the first speaker to Larry is longer than the path from the second speaker to Larry.
Now, to find the wavelengths that result in constructive and destructive interference, we can use the formula:
Path Difference = n * λ (for constructive interference)
Path Difference = (n + 1/2) * λ (for destructive interference)
Where:
n = integer representing the order of the interference
λ = wavelength
For constructive interference:
Δd = n * λ
For destructive interference:
Δd = (n + 1/2) * λ
Since we want Moe and Curly to hear very little sound, we can consider the case of destructive interference.
Let's calculate the two lowest frequencies that result in destructive interference for Larry.
Frequency (f) can be calculated using the formula:
f = c / λ
Where:
c = speed of sound (approximately 343 m/s at room temperature)
For the first frequency (n = 0):
Δd = (0 + 1/2) * λ
-2.20 m = (1/2) * λ
λ = -2.20 m / (1/2)
λ = -4.40 m
f1 = c / λ
= 343 m/s / (-4.40 m)
≈ -77.95 Hz
For the second frequency (n = 1):
Δd = (1 + 1/2) * λ
-2.20 m = (3/2) * λ
λ = -2.20 m / (3/2)
λ ≈ -1.473 m
f2 = c / λ
= 343 m/s / (-1.473 m)
≈ -233.05 Hz
Note: The negative frequencies indicate that the waves are reflected and interfering with each other.
Therefore, the two lowest frequencies that allow Larry to hear a loud tone while Moe and Curly hear very little are approximately -77.95 Hz and -233.05 Hz.
To find the two lowest frequencies that allow Larry to hear a loud tone while Moe and Curly hear very little, we need to consider the concept of constructive and destructive interference.
Constructive interference occurs when two waves are in phase and combine to create a wave with a larger amplitude. In this case, Larry will hear a loud tone when the waves from the two speakers constructively interfere at his position.
On the other hand, destructive interference occurs when two waves are out of phase and combine to create a wave with a smaller or zero amplitude. This means that Moe and Curly will hear very little or no sound when the waves from the two speakers destructively interfere at their positions.
Here's how we can approach the problem:
1. Determine the path difference between the two speakers to Larry.
- The path difference is the extra distance one wave travels compared to the other due to the spacing between the speakers.
- In this case, the path difference is equal to the spacing between the speakers, which is 0.800m.
2. Calculate the wavelength corresponding to the path difference.
- The path difference equals a whole number of wavelengths (constructive interference) or half a wavelength (destructive interference).
- Since the path difference is equal to the spacing between the speakers, the wavelength corresponding to constructive interference is given by λ = 2d = 2 * 0.800m = 1.600m.
- The wavelength corresponding to destructive interference is given by λ = 2 * d / 2 = d = 0.800m.
3. Find the frequencies corresponding to each wavelength.
- The speed of sound in air is approximately 343 m/s.
- Frequency (f) is related to wavelength (λ) by the formula: f = v / λ, where v is the speed of sound.
- For the wavelength corresponding to constructive interference, the frequency is f = 343 m/s / 1.600m = 214.4 Hz (rounded to one decimal place).
- For the wavelength corresponding to destructive interference, the frequency is f = 343 m/s / 0.800m = 428.8 Hz (rounded to one decimal place).
Therefore, the two lowest frequencies that allow Larry to hear a loud tone while Moe and Curly hear very little are approximately 214.4 Hz and 428.8 Hz.