-Suppose a hi-speed train is traveling down a long level section of track at a constant velocity, v. It is a windless day. The engineer at the engine decides to blow the whistle at some particular time. An observer in the caboose, and another observer on the train station platform may feel compelled to ask, what is the speed of the train. Can either or both find the speed of the train based only on the blast (sound) of the train whistle?

-The observer on the caboose might construct her algebra problem like this; she knows the distance, D, from the whistle to the caboose from the known train specifications. But since she also knows that the train is moving with the constant velocity v, the train must meet the sound pulse from the whistle somewhere within the distance between the caboose and the whistle (the whistle and the caboose are at rest relative to each other, hopefully). So she sets up her algebraic equation as follows:

:Envy: c_st = D - vt

Where c_s, is the speed of sound in air, and both times, t, are equal. Rearranging this equation:

D = c_st + vt

D = t(c_s + v)

[D /t] = (c_s + v)

[D/t] - c_s = v

-So, she imagines if she sends a very fast signal (such as a light signal - virtually instantaneous) to the whistle, starts a clock by her side when she sends the signal, then measures the time until she hears the whistle blast. She should be able to find the velocity of the train, v, from the above equation.

-On the other hand, the observer on the platform hears this sound pulse from the whistle at the same time. By starting a clock as he hears the first wave front, and then timing the time difference of the arrival of the second sound wave front, he can find the wavelength of the sound as the train approaches him at that instant:

:Envy: λ_f = c_st

But by Doppler, the wavelength in front of the moving sound source also equals:

:Envy: λ_f = (c_s - v) / ƒ₀

He knows the frequency, ƒ₀, of the whistle from the train specifications, so by substitution, he can solve for the velocity of the train, v:

:Envy: v = c_s - [(λ_f)(ƒ₀)]

-If the two observers measure the same whistle blast simultaneously, then will they find the same value for v, the velocity of the train?

http://en.wikipedia..../Doppler_effect