Noise-cancelling headphones are an application of destructive interference. Each side of the
headphones uses a microphone to pick up noise, delays it slightly, then rebroadcasts the noise
next to your ear where it can interfere with the incoming sound wave of the noise. Suppose you
are sitting 1.8 m from an annoying, 110 Hz buzzing sound. What is the minimum headphone
delay, in ms, that will cancel this noise?
To cancel the noise, the headphones need to create a sound wave that is 180 degrees out of phase with the incoming sound wave.
The minimum delay needed for destructive interference can be calculated using the formula:
Δt = (1/f) * (d/c)
Where:
Δt = Minimum delay (in seconds)
f = Frequency of the sound wave (in Hz)
d = Distance between the headphones and the source of the sound (in meters)
c = Speed of sound in air (approximately 343 m/s)
Plugging in the given values:
f = 110 Hz
d = 1.8 m
c = 343 m/s
Δt = (1/110) * (1.8/343)
Δt ≈ 0.00407 seconds
To convert the delay to milliseconds:
Δt_milliseconds = Δt * 1000
Δt_milliseconds ≈ 4.07 milliseconds
Therefore, the minimum headphone delay needed to cancel the noise is approximately 4.07 milliseconds.