Two speakers, one directly behind the other, are each generating a 406-Hz sound wave. What is the smallest separation distance between the speakers that will produce destructive interference at a listener standing in front of them? Take the speed of sound to be 344 m/s.
Well, here's a joke for you while we tackle the question: Why don't scientists trust atoms? Because they make up everything!
Now, let's get into the question! To determine the smallest separation distance between the speakers that will produce destructive interference, we can use the formula for destructive interference:
Δx = λ/2
Where Δx is the separation distance, and λ is the wavelength. Since we're given the frequency (406 Hz) and the speed of sound (344 m/s), we can find the wavelength using the formula:
λ = v/f
So, λ = 344 m/s / 406 Hz = 0.847 meters.
Now, let's plug that value back into our original formula to find the smallest separation distance:
Δx = 0.847 meters / 2 = 0.4235 meters.
Therefore, the smallest separation distance between the speakers that will produce destructive interference is approximately 0.4235 meters. But remember, always be careful not to trip over any punchlines!
To determine the smallest separation distance between the speakers that will produce destructive interference at the listener, we need to find the condition for destructive interference.
Destructive interference occurs when two waves are perfectly out of phase. In other words, the crests of one wave align with the troughs of the other wave. This leads to a cancellation of the sound.
The formula for destructive interference in terms of the wavelength (λ), the speed of sound (v), and the path difference (Δx) is:
Δx = (n + 0.5) * λ / 2
Where:
- Δx is the path difference between the two speakers
- n is an integer representing the order of the destructive interference (n = 0, 1, 2, ...)
- λ is the wavelength of the sound wave
- v is the speed of sound
We can rearrange this formula to solve for λ:
λ = 2 * (Δx / (n + 0.5))
Given that the frequency (f) of the sound wave is 406 Hz, we can calculate the wavelength (λ) using the formula:
λ = v / f
Substituting the values, we get:
λ = 344 m/s / 406 Hz
Now we can substitute this value of λ back into the equation for Δx:
Δx = (n + 0.5) * λ / 2
We are interested in the smallest value of Δx that produces destructive interference at the listener, so we can set n = 0:
Δx = (0 + 0.5) * λ / 2
Simplifying further:
Δx = 0.5 * λ / 2
Now we can substitute the value of λ we calculated earlier:
Δx = 0.5 * (344 m/s / 406 Hz) / 2
Calculating this expression:
Δx ≈ 0.212 m
Therefore, the smallest separation distance between the two speakers that will produce destructive interference at a listener standing in front of them is approximately 0.212 meters.
To determine the smallest separation distance between the speakers that will produce destructive interference at a listener standing in front of them, we need to consider the concept of interference between sound waves.
In this scenario, the two speakers are generating sound waves of the same frequency (406 Hz). When these waves combine, they can either reinforce each other (constructive interference) or cancel each other out (destructive interference). The condition for destructive interference to occur is when the path difference between the two waves is equal to an odd multiple of half the wavelength.
To find the wavelength (λ) of the sound wave, we can use the formula:
λ = v / f
where v is the speed of sound (344 m/s) and f is the frequency (406 Hz).
Plugging in the values, we get:
λ = 344 m/s / 406 Hz
Now that we know the wavelength, let's consider the path difference. The path difference is the difference in distance traveled by the sound waves from each speaker to the listener. For destructive interference to occur, the path difference must be equal to an odd multiple of half the wavelength.
Let's assume the separation distance between the speakers is represented by d. The path difference can be calculated as:
path difference (Δd) = d - λ/2
For destructive interference, the path difference should be equal to an odd multiple of half the wavelength. So, we can write the equation:
Δd = (2n + 1) * λ/2
where n is an odd integer.
Since we're interested in the smallest separation distance, we'll consider the case when n = 0. Substituting n = 0 into the equation, we get:
Δd = (2(0) + 1) * λ /2
Δd = λ /2
Therefore, the smallest separation distance (d) between the speakers that will produce destructive interference at the listener is equal to half the wavelength (λ/2).
To get the numerical value, we need to substitute the calculated value of λ (wavelength):
d = λ / 2 = (344 m/s / 406 Hz) / 2
Calculating this expression, we find the smallest separation distance:
d ≈ 0.423 m or 42.3 cm
So, the minimum separation distance between the speakers should be approximately 42.3 cm to produce destructive interference at a listener standing in front of them.