Question

At distance $r$ from a point source with power $P_{av}$, the wave intensity is $I=P_{av}/4\pi r^2$. prove that a point source with power $P_{av}$ moving with constant speed $v_S$. the wave intensity is
$I=\frac{P_{av}}{4\pi r^2}\left(\frac{v-v_S}{v}\right)$

Answer 1

We know that $P_{avg}=\frac{Energy}{time}=\frac{E}{T}$

Now, $f=\frac{1}{T}$

And as the source is moving, the frequency observed at that point will be

$f^{\prime }=f\left(\frac{v}{v-v_s}\right)⟹T^{\prime }=\frac{1}{f^{\prime }}$

$⟹T^{\prime }=T\left(\frac{v-v_s}{v}\right)$

So the new average power reaching that point $P_{avg}^{\prime }=\frac{E}{T^{\prime }}⟹P_{avg}=\frac{P_{avg}(v-v_s)}{v}$

Therefore the Intensity $I=\frac{P_{avg}}{4\pi r^2}\left(\frac{v-v_s}{v}\right)$