Interference — Question 8

Question 8

A fixed cancellation delay has a finite useful bandwidth

Two paths deliver equal pressure amplitudes $P$. Their relative transfer phase is $2\pi f\tau+\phi_0$, with fixed delay $\tau=2.0\,\mathrm{ms}$. Choose $\phi_0$ so the paths cancel at $f_0=500\,\mathrm{Hz}$.

1. Find the residual pressure amplitude at $f=f_0+\Delta f$.

2. Find the largest $|\Delta f|$ in the band around $f_0$ for which the residual amplitude is at most $0.10P$.

Question 8 - Solution

A fixed cancellation delay has a finite useful bandwidth

(a) Method and calculation

The chosen phase obeys $2\pi f_0\tau+\phi_0=\pi$ modulo $2\pi$. At the offset frequency,$$\delta=\pi+2\pi\Delta f\tau,\quad P_{\mathrm{res}}=2P|\cos(\delta/2)|=2P|\sin(\pi\Delta f\tau)|.$$

(b) Interpretation and further calculation

In the central band the sine magnitude increases monotonically. Therefore$$|\sin(\pi\Delta f\tau)|\le0.05,\quad |\Delta f|\le\frac{\sin^{-1}(0.05)}{\pi(0.002)}=7.961\,\mathrm{Hz}.$$A longer mismatch delay narrows the useful band. The approximate bound $0.05/(\pi\tau)$ differs very little here.

Physical check

At $\Delta f=0$ the residual vanishes. Additional remote cancellation bands exist, but the requested bound is the contiguous band around $f_0$.