Ever want to hear or see beat frequencies? Here you go!
WATCH THE VIDEOS
Hearing Beats
Seeing Beats
What Does Music Look Like |
Standing Waves and Strobe |
Wave Interference |
- Sound and Music
- Frequencies
- Interference
Theory:
Beat frequencies occur when the waves from two waves sources, each with different frequencies, interfere with each other. This produces a single tone whose volume oscillates between loud and soft. This tone that is constantly changing volume is what is referred to a beats.
In order to find these beats, consider two waves interfering. Assume their amplitudes are the same and that there is no phase difference. Thus the waves can be expressed as:
x_{1} = A cos (ω_{1}t)
x_{2} = A cos (ω_{2}t)
where A is the amplitude, ω is the angular frequency, t is time, and the subscripts 1 and 2 are used to differentiate between the two waves. Add the two waves together:
x = x_{1} + x_{2}
x = A cos(ω_{1}t) + A cos(ω_{2}t)
x = A [cos(ω_{1}t) + cos(ω_{2}t)]
To simplify, use the trig identity:
cos α + cos β = 2 cos[(α - β) / 2] cos[(α + β) / 2]
Thus we get:
x = 2A cos[(ω_{1} - ω_{2})t / 2] cos[(ω_{1} + ω_{2})t / 2]
If we let (ω_{1} + ω_{2}) / 2 = ω_{avg} (the average angular frequency) and let (ω_{1} - ω_{2}) / 2 = ω_{mod} (the modulation frequency), we can simplify to:
x = 2A cos(ω_{mod}t) cos(ω_{avg}t)
This formula for the addition of two waves is true for any values of ω_{1} and ω_{2}, but for beat frequencies we are looking at a certain case. For beat frequencies to occur, the two angular frequencies must be close in value. In other words:
|ω_{1} - ω_{2}| << ω_{1} + ω_{2}
This means that ω_{avg} is similar in value to ω_{1} or ω_{2}, while ω_{mod} is very small, close to zero. This means that the cos(ω_{mod}t) term is oscillating very slowly. Together with the 2A term, 2A cos(ω_{mod}t) provides the slowly changing amplitude for the cos(ω_{avg}t) function.
As it turns out, the beat frequency is twice the modulation frequency, f_{mod}. f_{mod} is equal to ω_{mod }divided by 2π. Thus:
where f_{1} and f_{2} are the frequencies of the two waves sources. Thus, the beat frequency is just the difference in the frequencies of the two wave sources.
Apparatus:
- Hearing Beats: Two tuning forks of different (but close) frequencies. In the video a tuning fork of 250 Hz and one of 238 Hz were used.
- Seeing Beats: Two tunable speakers and an oscilloscope attached to a microphone
Procedure:
- Set up the two tuning forks near each other and strike them one at a time.
- Listen for the beat frequency. It will be a single tone that is alternating between being loud and soft.
- Use an oscilloscope to "see" the beats