Derivation of Relativisitic Doppler effect with angle

[Arman777]

Homework Statement

Derive the formula for the Doppler effect for a receiver traveling at an angle theta away from a planar source

Homework Equations

The Attempt at a Solution

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I thought that we can assume that the wavelength has two components $λ_x$ and $λ_y$ where

$λ_x = ct+vtcosθ$ and I am not sure about $λ_y$. Maybe $λ_y = \sqrt {(ct)^2+ (vtsinθ)^2)}$

I am not sure how to start or approach please help.

Note : I cannot use wave equation.

 

[PeroK]

Are you going to model light as a wave or as massless particles, with energy and momentum?

 

[Arman777]

Are you going to model light as a wave or as massless particles, with energy and momentum?

I am not sure actually. Why its important ?. We can't use four vectors or etc.. we didnt learn them.. We can use basic stuff like lorentz transformation only actually.

 

[PeroK]

I am not sure actually. Why its important ?. We can't use four vectors or etc.. we didnt learn them.. We can use basic stuff like lorentz transformation only actually.

Okay, so you can't use the energy-momentum transformation. What about expressing light as a plane wave?

 

[Orodruin]

I suggest the following:

• Write down the phase function of the wave. You can assume a plane wave moving in the $x$-direction for this purpose.
• Write down an expression for the world line of the observer.
• For a small change in $t$, determine the change $d\phi$ in the phase function along the world line of the observer.
• Determine the proper time $ds$ passed for the observer in the same small time difference.
• The frequency observed by the observer is $d\phi/ds$.
• You can find the original frequency by doing the same for the emitter.

 

[Arman777]

I suggest the following:

• Write down the phase function of the wave. You can assume a plane wave moving in the $x$-direction for this purpose.
• Write down an expression for the world line of the observer.
• For a small change in $t$, determine the change $d\phi$ in the phase function along the world line of the observer.
• Determine the proper time $ds$ passed for the observer in the same small time difference.
• The frequency observed by the observer is $d\phi/ds$.
• You can find the original frequency by doing the same for the emitter.

Thats kind of hard for me to do. I can try step by step tho but I am not sure I can find it. Is there a simpler way ? Or is there a way without writing plane-wave equation ? . I find a online derviation and it uses planes waves and lorentz transformation. Probably this is what you are trying to describe.

I

 

[rude man]

The wording of the problem would seem to allow starting with the expression for the relativistic Doppler shift head-on. Then just figure out the effect of theta.

 

[Arman777]

The wording of the problem would seem to allow starting with the expression for the relativistic Doppler shift head-on. Then just figure out the effect of theta.

How..

 

[rude man]

How..

First, you realize they're talking about light waves, right? I mean, couldn't be sound waves at relativistic speeds.

So how about using the relativistic expression for straight-back light travel (not at an angle to the source-receiver line), then considering how the effective rate of travel as seen by the receiver is impacted by an angle other than zero? Looks like plain old trig to me - but that's just me.