Beats

Interference with slightly different frequencies

Beats occur when two waves with nearly equal frequencies overlap. Instead of producing a steady tone, the combined sound grows louder and softer periodically. This pulsing is called beating.

For sound, beats are easy to hear when two tuning forks or instruments play frequencies close together. If one is slightly out of tune, the loudness rises and falls at a rate related to the frequency difference.

Adding two cosine waves

Consider two waves at the same location with equal amplitude:

$y_1=Acos(omega_1 t)$

$y_2=Acos(omega_2 t)$

The total is

$y=y_1+y_2=Acos(omega_1 t)+Acos(omega_2 t)$

Using the identity

ight)cosleft(rac{a+b}{2} ight)$$ we get $$y=2Acosleft(rac{omega_1-omega_2}{2}t ight)cosleft(rac{omega_1+omega_2}{2}t ight)$$ This expression shows a fast oscillation multiplied by a slowly varying envelope. ## Beat frequency The rapid oscillation occurs near the average angular frequency: $$omega_{avg}=rac{omega_1+omega_2}{2}$$ The amplitude envelope varies with the difference frequency. In terms of ordinary frequencies, the beat frequency is $$f_{beat}=|f_1-f_2|$$ This is the number of loudness maxima per second. For example, if two tuning forks have frequencies $440 Hz$ and $444 Hz$, the beat frequency is $$4 Hz$$ meaning the sound grows loud four times per second. ## Why the beat frequency is not half the difference The envelope factor contains $$cosleft(rac{omega_1-omega_2}{2}t ight)$$ but loudness depends on the magnitude of the amplitude, and both positive and negative envelope peaks sound loud. Therefore the audible beat frequency corresponds to the full frequency difference $|f_1-f_2|$. ## Tuning instruments Musicians use beats to tune instruments. If two notes are intended to match but beats are heard, their frequencies differ. As the instrument is adjusted closer to the reference pitch, the beats slow down. When the beats disappear, the frequencies are matched. This method is extremely sensitive because humans can detect slow variations in loudness. ## Beats beyond sound Beats are not limited to sound. They appear in radio signals, optics, coupled oscillators, and quantum systems. Any time two nearby frequencies are superposed, amplitude modulation can occur. In communications, beating and modulation are related ideas. Combining frequencies can shift information into different frequency ranges. ## Wave packets Beats also introduce the idea of a wave packet. When waves of different frequencies combine, the result can have an envelope that moves differently from the individual wave crests. This leads to the distinction between phase velocity and group velocity in more advanced wave physics. The beat pattern is an early example of how superposition can create larger-scale structures from simple sinusoidal waves. ## The big idea Beats are caused by interference between waves of slightly different frequencies. The combined wave has a fast oscillation and a slowly varying amplitude envelope. The beat frequency is the absolute difference between the two frequencies. Beats are important in music, tuning, signal processing, and the broader study of wave packets.