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Re: Doppler Effect (constant acceleration)



Regarding Wilson's question:

Have any one heard of an easy way to measure the change in the
frequency of a sound source when the source is moving linearly with
constant acceleration toward a stationary observer? As you know, the
Doppler effect describes, among other scenarios, the change in
perceived frequency of a sound source when the object is moving with
constant speed toward a stationary observer. A colleague and I were
having a discussion about that. Since he is a physicist and I am a
science educator, I am almost sure he is correct when he argues it
cannot be done. However, common sense tells me that if a sound-
emitting object is moving with constant acceleration, the perceived
sound frequency should change in a predictable way. This case should
not be confused with a common demonstration of Doppler effect using
a rotating sound source.

Suppose that f_s is the monochromatic source frequency, and that v_0
is the closing speed of the source toward the receiver at time t = 0,
and that a is the constant acceleration of the source toward the
stationary observer. Then the observed 'instantaneous' frequency
at time t is given by f(t) = f_s/(1 - (v_0 + a*t)/c) .

Of course since the signal the observer receives is no longer a
monochromatic sine-wave signal, it has a complicated Fourier
decompostion in terms of real frequencies, but that signal can still
be represented by a frequency modulated sine wave such that the
carrier frequency is shifted as a function of time according to the
formula above.

David Bowman