Question

Noise-canceling 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, 200 Hz buzzing sound.
What is the minimum headphone delay, in ms, that will cancel this noise?

Answer 1

Answer:

2.5 ms

Explanation:

v = Speed of sound in air = 343 m/s

f = Frequency = 200 Hz

Wavelength is given by

$\lambda=\dfrac{v}{f}\\\Rightarrow \lambda=\dfrac{343}{200}\\\Rightarrow \lambda=1.715\ m$

In the case of destructive interference, path difference is given by

$x=\dfrac{\lambda}{2}\\\Rightarrow x=\dfrac{1.715}{2}\\\Rightarrow x=0.8575\ m$

Delay is givenn by

$t=\dfrac{x}{v}\\\Rightarrow t=\dfrac{0.8575}{343}\\\Rightarrow t=0.0025\ s$

The minimum headphone delay, that will cancel this noise is 2.5 ms