Two speakers, one directly behind the other, are each generating a 579-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.
To find the smallest separation distance between the speakers that will produce destructive interference at a listener, we need to consider the phase difference between the sound waves generated by the speakers.
Destructive interference occurs when two waves are out of phase and their troughs and peaks align. In this case, we want the waves to be exactly half a wavelength out of phase.
The formula for the wavelength of a sound wave is given by:
λ = v / f
where λ is the wavelength, v is the speed of sound, and f is the frequency.
For the first speaker, the wavelength (λ₁) is:
λ₁ = v / f₁
For the second speaker, the wavelength (λ₂) is:
λ₂ = v / f₂
Since the speakers are generating the same frequency (579 Hz), their wavelengths are the same.
Therefore,
λ₁ = λ₂
v / f₁ = v / f₂
Since v is constant,
f₁ = f₂
So, the frequency of the sound waves generated by each speaker is the same.
Now, let's consider the path difference between the waves reaching the listener. The path difference depends on the distance between the speakers (d) and the angle at which the waves reach the listener.
To achieve destructive interference, the path difference must be half a wavelength:
Δx = λ/2
Notice that Δx is equal to the horizontal distance between the listener and the nth nodal line (i.e., point of destructive interference).
In this case, since the speakers are one behind the other, the angle at which the waves reach the listener is very close to zero. Therefore, we can consider the path difference as the horizontal separation distance between the speakers.
Therefore, we have:
Δx = d
Setting Δx equal to λ/2, we can solve for d:
d = λ/2
Substituting the value of λ:
d = v / (2f)
Now we can substitute the given values:
v = 344 m/s (speed of sound)
f = 579 Hz (frequency of the sound wave)
d = 344 / (2 * 579)
d ≈ 0.297 m
So, the smallest separation distance between the speakers that will produce destructive interference at the listener is approximately 0.297 meters.