Undergrad Classical Aberration Formula: Understanding & Application

[Kairos]

I believe I have understood the formula of aberration of light $\tan \theta' = \dfrac{\sin \theta}{\beta + \cos \theta} \sqrt{1-\beta^{2}}$

but I wonder if the non-relativistic formula $\tan \theta' = \dfrac{\sin \theta}{\beta + \cos \theta}$ has a physical relevance. Does this formula apply to a real physical situation (for sound ?) or is it just an approximation for small $\beta$?

 

[Orodruin]

Did you ever drive a car in rain? Did the rain seem to be coming from ahead?

 

[Kairos]

Yes I did. But as the point of emission of the rain is not well defined, I have difficulty in defining $\sin \theta$ and $\sin \theta'$

 

[Orodruin]

Yes I did. But as the point of emission of the rain is not well defined, I have difficulty in defining $\sin \theta$ and $\sin \theta'$

The emission point is irrelevant. The only relevant thing is the angle of the rain relative to the velocity of the car.

 

[Kairos]

Does this mean that this formula is valid for an emission by a plane but not for a point source?

 

[ergospherical]

But as the point of emission of the rain is not well defined, I have difficulty in defining $\sin \theta$ and $\sin \theta'$

$\theta$ and $\theta'$ have nothing a priori to do with the emission point, they're related to the angle between the velocity of the particle and the coordinate axes i.e. $\mathbf{v} = (v\cos{\theta}, v\sin{\theta})$ and $\mathbf{v}' = (v'\cos{\theta'}, v'\sin{\theta'})$.

 

[Kairos]

thank you

 

[Meir Achuz]

Stellar aberration is so small that the the NR expression, derived and observed in 1729 by James Bradley,
is sufficient. A simpler measurable result is $\theta -\theta'=\beta \sin\theta.$