Two loudspeakers in a 20°C room emit 629 Hz sound waves along the x-axis.
(a) If the speakers are in phase, what is the smallest distance between the speakers for which the interference of the sound waves is destructive?
(b) If the speakers are out of phase, what is the smallest distance between the speakers for which the interference of the sound waves is constructive?
Nevermind, got it, thanks anyway.
To solve this question, we need to use the concept of interference of sound waves. Interference occurs when two or more waves combine and interact with each other.
(a) Destructive interference occurs when two waves are completely out of phase, meaning their crests and troughs align. In this case, we need to find the smallest distance for destructive interference.
The formula to calculate the distance for destructive interference is given by:
d = λ/2
Where:
d = distance between the speakers
λ = wavelength of the sound waves
To find the wavelength, we can use the formula:
λ = v/f
Where:
v = speed of sound in air (approximately 343 m/s at 20°C)
f = frequency of the sound waves (given as 629 Hz)
Substituting the values in the formula:
λ = 343/629
λ ≈ 0.546 m
Now, we can calculate the distance for destructive interference:
d = 0.546/2
d ≈ 0.273 m
Therefore, the smallest distance between the speakers for destructive interference is approximately 0.273 m.
(b) Constructive interference occurs when two waves are completely in phase, meaning their crests and troughs align. In this case, we need to find the smallest distance for constructive interference.
The formula to calculate the distance for constructive interference is given by:
d = λ/2
Since the frequency of the sound waves and the speed of sound in air remain the same, the wavelength remains the same too (0.546 m).
Now, we can calculate the distance for constructive interference:
d = 0.546/2
d = 0.273 m
Therefore, the smallest distance between the speakers for constructive interference is also approximately 0.273 m.
Note: The result is the same for both destructive and constructive interference because the frequency of the sound waves is the same (629 Hz) and does not affect the result.
To find the smallest distance between the speakers for destructive interference, we need to use the equation for destructive interference:
d = λ / 2
where d is the distance between the speakers and λ is the wavelength of the sound waves.
To find the wavelength, we can use the formula:
λ = v / f
where λ is the wavelength, v is the speed of sound, and f is the frequency of the sound wave.
Step 1: Convert temperature to Kelvin
The speed of sound depends on the temperature of the room. To convert the temperature from Celsius to Kelvin, use the formula:
T(K) = T(°C) + 273.15
Given: T(°C) = 20°C
T(K) = 20 + 273.15 = 293.15 K
Step 2: Calculate the speed of sound
The speed of sound in air can be approximated using the formula:
v = 331.5 m/s + 0.6 m/s · T(K)
Given: T(K) = 293.15 K
v = 331.5 m/s + 0.6 m/s · 293.15 K ≈ 331.5 m/s + 175.89 m/s ≈ 507.39 m/s
Step 3: Calculate the wavelength
Using the equation λ = v / f, we can find the wavelength. Given f = 629 Hz:
λ = 507.39 m/s / 629 Hz = 0.806 m
Step 4: Calculate the distance between the speakers for destructive interference
Using the equation d = λ / 2, we can find the distance between the speakers:
d = 0.806 m / 2 = 0.403 m
Therefore, the smallest distance between the speakers for destructive interference is approximately 0.403 meters.
To find the smallest distance between the speakers for constructive interference, we use the equation for constructive interference:
d = λ / 2
Using the same wavelength calculated earlier (0.806 m):
d = 0.806 m / 2 = 0.403 m
Therefore, the smallest distance between the speakers for constructive interference is approximately 0.403 meters.