Doppler Effect - General

General

In classical physics, where the speeds of source and the receiver relative to the medium are lower than the velocity of waves in the medium, the relationship between observed frequency f and emitted frequency f₀ is given by:

where

is the velocity of waves in the medium;

is the velocity of the receiver relative to the medium; positive if the receiver is moving towards the source;

is the velocity of the source relative to the medium; positive if the source is moving away from the receiver.

The frequency is decreased if either is moving away from the other.

The above formula works for sound waves if and only if the speeds of the source and receiver relative to the medium are slower than the speed of sound. See also Sonic boom.

The above formula assumes that the source is either directly approaching or receding from the observer. If the source approaches the observer at an angle (but still with a constant velocity), the observed frequency that is first heard is higher than the object's emitted frequency. Thereafter, there is a monotonic decrease in the observed frequency as it gets closer to the observer, through equality when it is closest to the observer, and a continued monotonic decrease as it recedes from the observer. When the observer is very close to the path of the object, the transition from high to low frequency is very abrupt. When the observer is far from the path of the object, the transition from high to low frequency is gradual.

In the limit where the speed of the wave is much greater than the relative speed of the source and observer (this is often the case with electromagnetic waves, e.g. light), the relationship between observed frequency f and emitted frequency f₀ is given by:

where

is the velocity of the source relative to the receiver: it is positive when the source and the receiver are moving farther apart.

is the speed of wave (e.g. 3×108 m/s for electromagnetic waves travelling in a vacuum)

is the wavelength of the transmitted wave in the reference frame of the source.

These two equations are only accurate to a first order approximation. However, they work reasonably well when the speed between the source and receiver is slow relative to the speed of the waves involved and the distance between the source and receiver is large relative to the wavelength of the waves. If either of these two approximations are violated, the formulae are no longer accurate.