Question

Prove the Doppler Effect: (a)detector moving, source stationary. (b) source moving, detector stationary. (c) general Doppler Effect equation.

Answer 1

Consider a stationary source of sound broadcasting a single frequency sound wave. You are the observer of the sound wave, and you are also stationary.

The usual relationship between frequency, speed, and wavelength is:

f = v/λ

v represents the speed of sound through the medium.

Let's say you, the observer, now move toward the source with velocity v_O. You encounter more waves per unit time than you did before. Relative to you, the waves travel at a higher speed:

v^/ = v + v_O

The frequency of the waves you detect is higher, and is given by:

f^/ = v^//λ = (v + v_O) /λ

If you moved away from the source the observed frequency is lower. In general the observed frequency when the observer moves is:

f^/ = (v +/- v_O) /λ = (v +/- v_O)* f/v = f * (v +/- v_O)/v

Use the first sign (+) when the detector moves toward the source and the second sign (-) when the detector moves away.

(b)

What happens when the source of the waves moves toward you, a stationary observer? Again, you encounter more waves per unit time than you did before so the frequency is shifted up. This time, though, the shift occurs because the wavelength has been lowered by the movement of the source.

When nothing moves the wavelength is equal to vT, where T is the period, or v/f, because T = 1/f. When the source moves at speed v_s, the wavelength is different by the distance traveled by the source in one period:

Change in wavelength = v_sT = v_s/f

The effective wavelength is λ ^/ = v/f -/+ v_s/f = [v -/+ v_s]/f

Use the first sign (-) when the source moves toward the observer, and the second sign (+) when it moves away.

The detected frequency is:

f^/ = v/λ ^/ = f v/[v -/+ v_s ]   

(c)

In some situations both the source and the observer move. We can write out a general Doppler equation for the observed frequency by simply combining the previous results.

The general equation accounting for any motion is:

f^/ = f (v +/- v_O) / (v -/+ v_s )

For both sets of signs use the first sign when the motion is toward the other thing, and the second sign when the motion is away.