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tssuser
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I'm confused by problem 2.31 in mathematical tools for physics.
Problem:
2.31 The Doppler effect for sound with a moving source and for a moving observer have different formulas. The Doppler
effect for light, including relativistic effects is different still. Show that for low speeds they are all about the same.
[itex]f' = f \frac{v - v_0}{v}[/itex], [itex]f' = f \frac{v}{v+v_s}[/itex], [itex]f' = f \sqrt{\frac{1-v/c}{1+v/c}}[/itex]
The symbols have various meanings: v is the speed of sound in the first two, with the other terms being the velocity
of the observer and the velocity of the source. In the third equation c is the speed of light and v is the velocity of the
observer. And no, 1 = 1 isn't good enough; you should get these at least to first order in the speed.Solution:
From the selected solutions:
[itex]f' = f(1-v_0/v)[/itex], [itex]f' = f(1-v_s/v)[/itex], [itex]f'=f(1-v/c)[/itex]
Question:
Clearly I'm supposed to do a tailor expansion of something, but I'm unsure of which part of the original differential equation I'm supposed to expand. Also, whichever part I do expand I end up with a different result than the given solution, which makes me think I'm interpreting the equation wrong. My interpretation is:
[itex] f'(x) = \frac{v - v_0}{v} f(x)[/itex]
Thanks for any help clearing this up.
Problem:
2.31 The Doppler effect for sound with a moving source and for a moving observer have different formulas. The Doppler
effect for light, including relativistic effects is different still. Show that for low speeds they are all about the same.
[itex]f' = f \frac{v - v_0}{v}[/itex], [itex]f' = f \frac{v}{v+v_s}[/itex], [itex]f' = f \sqrt{\frac{1-v/c}{1+v/c}}[/itex]
The symbols have various meanings: v is the speed of sound in the first two, with the other terms being the velocity
of the observer and the velocity of the source. In the third equation c is the speed of light and v is the velocity of the
observer. And no, 1 = 1 isn't good enough; you should get these at least to first order in the speed.Solution:
From the selected solutions:
[itex]f' = f(1-v_0/v)[/itex], [itex]f' = f(1-v_s/v)[/itex], [itex]f'=f(1-v/c)[/itex]
Question:
Clearly I'm supposed to do a tailor expansion of something, but I'm unsure of which part of the original differential equation I'm supposed to expand. Also, whichever part I do expand I end up with a different result than the given solution, which makes me think I'm interpreting the equation wrong. My interpretation is:
[itex] f'(x) = \frac{v - v_0}{v} f(x)[/itex]
Thanks for any help clearing this up.