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,

110 Hz buzzing sound. What is the minimum headphone delay, in

ms, that will cancel this noise?

Short answer

The minimum headphone delay that will cancel this noise will be$0.45ms.$

Step-by-step solution

Given information

The frequency of the buzzing sound is, $f=110Hz.$

The maximum destructive interference occurs in noise cancellation headphones.

Determination of delayed time

For the destructive interference to be happen, there must be a phase difference of $\mathrm{\pi }$between the waves that corresponds to a time period of $\frac{T}{2}$.

Thus, the delayed time for which the noise will be cancelled, should be,

$$\begin{gathered} ∆t=\frac{T}{2} \\ ∆t=\frac{1}{2f}\left[\because f=\frac{1}{T}\right] \\ ∆t=\frac{1}{2\times 110Hz} \\ ∆t=0.0045s \\ ∆t=0.45ms \end{gathered}$$