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Doppler Effect

The Doppler effect, named after Austrian physicist J. C. Doppler who first described it for sound in 1842, states that waves emitted from a source moving toward an observer are squeezed; i.e. the wave's wavelength is decreased and frequency is increased, as shown in the schematic below. Conversely, waves emitted from a source moving away from an observer are stretched; i.e. the wave's wavelength is increased and frequency is decreased. The waves can be acoustic waves or electro-magnetic radiation.

Doppler Effect
Doppler Formula
Consider a monochromatic (single frequency) light source, such as a laser, with frequency f and wavelength l. The speed of light c is related to the frequency and wavelength by,

c = l · f

Assume that the light source is a distance d from the observer. If the light source and the observer are both stationary, the light wave takes n cycles to propagate from the source to the observer,

n = d / l

Suppose now the light source moves toward the observer at a velocity v. The distance required to propagate to the observer shrinks from d to,

(1 - v / c) · d

while the number n remains the same. Thus, the wavelength is compressed by a ratio of v / c and the observed wavelength lr is,

lr = (1 - v / c) l

Since the speed of light is constant,

c = l · f = lr · fr

the observed frequency is found to be,

If this observed frequency fr can be measured and/or compared to the at rest frequency f, the velocity of the light source can be obtained.

Irradiance of Two Light Beams

Consider a monochromatic light source, such as a laser, that has frequency f and wavelength l, that sends a light beam to illuminate a moving target. Further assume that the target moves toward the observer at the velocity v. According to the Doppler effect, the wavelength of the reflected light is compressed to lr and the frequency increases to fr.

A light detector can collect both incident and reflected light. Mathematically, the sum of these two light beams is,

where A is the amplitude of the incident light and Ar is the amplitude of the reflected light.

The irradiance is found to be,

Simplifying the third term in the last equation above (via a trigonometric product rule), we have,

Suppose the relative velocity v is much less than the speed of light (v « c). The observed frequency fr does not change very much from the original frequency f. As a result, the final term in the equation above is the only low frequency component. All of the other terms in the equation are at a frequency of f, fr, or higher.

By adding a low pass filter to the measurement, the frequency difference between the incident and reflected light beams can be obtained, allowing one to find the velocity of the target based on the Doppler effect. Such a low-pass filter is usually employed by Ultrasonic Doppler velocimeters.

In Laser Doppler anemometry, optical methods such as the interference of two laser beams are more preferable.