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Suppose there is a speaker emitting a steadily changing frequency sound while moving counterclockwise in a circle with radius of $r=1 \space m$. There is an observer listening to a sound with the same frequency located 2 meters from the outside of the circle, as seen in the figure below:

Gideon Bass. Created for Copyright 2016. All rights reserved.

The sound moves at a constant angular speed $\omega$.

You may assume the motion is slow enough that the Doppler Effect is insignificant, and that the speed of sound in air is 350 m/s.

The frequency of both speakers change together at a constant rate, such that the observer never hears a sound.

Which of the following could be an expression of the frequency as a function of time? Let $t=0$ be when $\theta=0$, i.e. when the speaker is at the opposite end of the circle relative to the observer.


$f = \cfrac{175}{\sqrt{2+\mbox{cos}^2(\omega t) +\mbox{sin}^2(\omega t)}}$


$f = \cfrac{350}{\sqrt{2+\mbox{cos}^2(\omega t) +\mbox{sin}^2(\omega t)}}$


$f = \cfrac{175}{\sqrt{\Big(2+\mbox{cos}(\omega t)\Big)^2 +\mbox{sin}^2(\omega t)}}$


$f = \cfrac{700}{\sqrt{\Big(2+\mbox{cos}(\theta )\Big)^2 +\mbox{sin}^2(\theta)}}$


There is no possible solution that would cancel the observed sound.

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