# Beats & Doppler Effect

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## Beats

The phenomenon of wavering of sound intensity when two waves of nearly same frequencies and amplitudes traveling in the same direction, are superimposed on each other is called beats

1. If we listen, a few minutes apart, two sounds of very close frequencies, say 256 Hz and 260 Hz, we will not be able to discriminate between them.
2. However, if both these sounds reach our ears simultaneously, what we hear is a sound of frequency 258 Hz, the average of the two combining frequencies.
3. In addition, we hear a striking variation in the intensity of sound — it increases and decreases in slow, wavering beats that repeat at a frequency of 4 Hz, the difference between the frequencies of two incoming sounds.

## Doppler Effect

The Doppler Effect is a change in the observed frequency of a wave when the source and the observer O move relative to the medium. For sound the observed frequency ν is given in terms of the source frequency v0 by

here v is the speed of sound through the medium, v0 is the velocity of observer relative to the medium, and vs is the source velocity relative to the medium. In using this formula, velocities in the direction OS should be treated as positive and those opposite to it should be taken to be negative.

If v, v0, vs, and vm are the velocities of sound, observer, source and medium respectively, then the apparent frequency

All velocities are taken positive in the source to observer (S → O) direction and negative in the opposite (O → S) direction.

Velocities v, v0, vs, and vm are in ms-1 and frequencies v and v’ in Hz.