Some exercises on relativistic doppler shift?

[Mr. Heretic]

My physics teacher has informed us that for our mock waves exam we are going to be given some relativistic questions on doppler shift, as an experiment by our loving masters.
We derived the Lorentz factor and some shift equations, but we haven't done anything else and I'd really like some practice applying them before the real thing, however I've been unsuccessful finding any exercises.

Could anyone point me in the right direction, or perhaps scan/photograph some questions they've had to do?
If the latter, answers under spoiler tags (or whatever they're called) would be appreciated too.

Note: The exam is fairly nigh (within 4 days of me posting this) sooner would be better than later, not to be demanding or anything. :-/

Note 2: I may end up writing myself some questions, then putting my working/answer up here seeking naught but confirmation.

 

[ghwellsjr]

Are you asking how to derive the Relativistic Doppler Factor from the Lorentz Transform or are you asking how to use the Relativistic Doppler Factor to solve a problem?

 

[Mr. Heretic]

I was bored so I went with my own problem (see Note 2).

But before I go into that, this isn't a particularly creative or difficult problem, so I'd just like to say I'm definitely still interested in more.
And I have an important question. What is the formula for moving-observer/stationary-source? Is it the same as the formula for moving source? I can't find it stated anywhere.


Problem goes as follows:

A 532.000 nm laser is mounted on the front of an experimental relativistic spacecraft , which then accelerates towards a distant, stationary observer until it reaches a velocity of 0.750000c.

a) What is the peak frequency observed?

b) Say next the laser is mounted on the back of the craft, and this time it accelerated in the opposite direction to the same top speed of 0.750000c, what is the peak wavelenth observed?

b. ii) How many nm longer would this peak be if the craft could and did achieve 0.990000c?


My answers:

a) 299792458/(532*10^-9)*√(1 +0.75)/√(1 -0.75) = 1.49093 x10^15 Hz

b) 532*√(1 +0.75)/√(1 -0.75) = 1407.54 nm

b. ii) 532*(√(1 +0.99)/√(1 -0.99) -√(1 +0.75)/√(1 -0.75)) = 6097.24 nm.

 

[Mr. Heretic]

Neither, ghwellsjr, I'm simply looking for examples of problems involving use of said factor/method to solve, such as above.

 

[ghwellsjr]

And I have an important question. What is the formula for moving-observer/stationary-source? Is it the same as the formula for moving source? I can't find it stated anywhere.

The only thing that matters is the relative velocity between the two observers. The effect is reciprocal so it doesn't matter if you use a frame to define one as stationary and the other as moving or vice versa. But if are using a frame in which they are both defined as moving then you can't just add or subtract their velocities, you have to use the relativistic velocity addition formula to calculate their relative velocity. And keep in mind that this simple Relativistic Doppler Formula only applies for two observers that are traveling along the same line. For transverse motion, it gets a lot more complicated.

And one more thing, you also have to consider the time it takes for the light to travel between the observers so if they are relatively at rest, separated by a great distance and then one of them accelerates, the change in frequency/wavelength will not immediately be observed.

I didn't work through your examples in detail to see if you did them right but it looks like you have a good understanding of what you are doing.

An excellent problem to solve using Relativistic Doppler is the famous Twin Paradox.

 

[Mr. Heretic]

Thank you, I had a feeling they might be reciprocal.
I'm fairly sure we won't go into the more complex stuff in the exam so it doesn't matter too much, but be I'll looking into relativistic velocity addition and the twin paradox.