Doppler effect

The Doppler effect is the apparent change in frequency or wavelength of a wave that is perceived by an observer moving relative to the source of the waves. For waves, such as sound waves, that propagate in a wave medium, the velocity of the observer and the source are reckoned relative to the medium in which the waves are transmitted. The total Doppler effect may therefore result from both motion of the source and motion of the observer. Each of these effects is analyzed separately.

Table of contents

1 Introduction 2 Applications 2.1 Everyday 2.2 Astronomy 2.3 Radar

Introduction

The effect was first analyzed by Christian Andreas Doppler in 1845. He then proceeded to test his analysis for sound waves by standing next to a rail line and listening to a car full of musicians as they approached him and after they passed him. He confirmed that sound's pitch was higher as the sound source approached him, and lower as the sound source receded from him, to the degree that he had predicted.

It is important to realize that the frequency of the sounds that the source emits does not actually change. To understand what happens, consider the following analogy. Someone throws one ball every second in your direction. Assume that balls travel with constant velocity. If the thrower is stationary, you will receive one ball every second. However, if he is moving towards you, you will receive more than that because there will be less spacing between the balls. The converse is true if the person is moving away from you. So it is actually the wavelength which is affected; as a consequence, the perceived frequency is also affected.

If the moving source is emitting waves with an actual frequency f₀, then an observer stationary relative to the medium detects waves with a frequency f given by:

,

(where v is the speed of the waves in the medium and v_s,r is the speed of the source with respect to the medium (positive if moving towards the observer, negative if moving away) radial to the observer.

A similar analysis for a moving observer and a stationary source yields the observed frequency (the observer's velocity being represented as v_o):

The first attempt to extend Doppler's analysis to light waves was soon made by Fizeau. In fact, light waves do not require a medium to propagate and the correct understanding of the Doppler effect for light requires the use of the Special Theory of Relativity. See relativistic Doppler effect.

Applications

Everyday

The siren on a passing emergency vehicle will start out higher than its stationary pitch, slide down as it passes, and continue lower than its stationary pitch as it receedes from the observer. John Dobson has described this effect:

"The reason the siren slides is because it doesn't hit you."

In other words, if the siren approached you directly, the pitch would remain constant (as v_s,r is only the radial component) until the vehicle hit you, and then immediately jump to a new lower pitch. The difference between the higher pitch and rest pitch would be the same as the lower pitch and rest pitch. Because the vehicle passes by you, the radial velocity does not remain constant, but instead varies as a function of the angle between your line of sight and the siren's velocity:

where v_s is the velocity of the object (source of waves) with respect to the medium, and is the angle between the object's forward velocity and the line of sight from the object to the observer.

Astronomy

The Doppler effect for light has been of great use in astronomy. It has been used to measure the speed at which stars and galaxies are approaching or receding from us, to detect that an apparently single star is, in fact, a close binary and even to measure the speed of rotation of stars and galaxies.

The use of the Doppler effect for light in astronomy depends on the fact that the spectra of stars are not continuous. They show absorption lines at well defined frequencies that are correlated with the energies required to excite electrons in various elements from one level to another. The Doppler effect in recognizable in that fact that the absorption lines are not always at the frequencies that are obtained from the spectrum of a stationary light source. Since blue light has a higher frequency than red light, the spectral lines from an approaching astronomical light source show a blueshift and those of receding sources show a redshift.

The measured Doppler effect for most remote galaxies shows them to be moving away from us. These data can be used to estimate the age of the universe (see redshift and Hubble's Law).

Radar

The Doppler effect is also used in some forms of radar to measure the velocity of detected objects. A radar beam is fired at a moving target - a car, for example, as radar is often used by police to detect speeding motorists - as it recedes from the radar source. Each successive wave has to travel further to reach the car, before being reflected and re-detected near the source. As each wave has to move further, the gap between each wave increases, increasing the wavelength. In some situations, the radar beam is fired at the moving car as it approaches, in which case each successive wave travels a lesser distance, decreasing the wavelength. In either situation, calculations from the Doppler effect accurately determine the car's velocity.