Interference — Question 3

Question 3

Motion between speakers changes the two path lengths oppositely

Two in-phase sources lie at $x=0$ and $x=d=2.0\,\mathrm m$. Consider $0<x<d$. Take $\lambda=0.80\,\mathrm m$ and, to isolate phase geometry, assume equal received amplitudes independent of $x$.

1. Find all positions of complete cancellation between the sources.

2. Find the spacing between neighboring cancellation points and explain the factor of two in the path change.

Question 3 - Solution

Motion between speakers changes the two path lengths oppositely

(a) Method and calculation

The distances are $r_1=x$ and $r_2=d-x$, so signed path difference is $\Delta r=d-2x$. For destructive interference,$$d-2x=(m+\tfrac12)\lambda,\qquad x=1.0-0.4(m+\tfrac12).$$Requiring $0<x<2$ gives $x=0.40,\ 0.80,\ 1.20,\ 1.60\,\mathrm m$. The endpoints are excluded.

(b) Interpretation and further calculation

Adjacent allowed values differ by$$|\Delta x|=\lambda/2=0.40\,\mathrm m.$$Moving the receiver by $dx$ increases one path by $dx$ and decreases the other by $dx$, so $d(\Delta r)=-2dx$. The midpoint has equal paths and is constructive, not a cancellation point.

Physical check

The listed points are symmetric about the midpoint. Real point-source spreading would prevent exact off-center cancellation unless amplitudes were compensated.