Beats
Beats
Striking two adjacent keys on a piano produces a warbling combination usually considered to be unpleasant. The superposition of two waves of similar but not identical frequencies is the culprit. Another example is often noticeable in jet aircraft, particularly the two-engine variety, while taxiing. The combined sound of the engines goes up and down in loudness. This varying loudness happens because the sound waves have similar but not identical frequencies. The discordant warbling of the piano and the fluctuating loudness of the jet engine noise are both due to alternately constructive and destructive interference as the two waves go in and out of phase. This figure illustrates this graphically.
Beats are produced by the superposition of two waves of slightly different frequencies but identical amplitudes. The waves alternate in time between constructive interference and destructive interference, giving the resulting wave a time-varying amplitude.
The wave resulting from the superposition of two similar-frequency waves has a frequency that is the average of the two. This wave fluctuates in amplitude, or beats, with a frequency called the beat frequency. We can determine the beat frequency by adding two waves together mathematically. Note that a wave can be represented at one point in space as
\(x=X\phantom{\rule{0.25em}{0ex}}\text{cos}(\cfrac{2\pi \phantom{\rule{0.25em}{0ex}}t}{T})=X\phantom{\rule{0.25em}{0ex}}\text{cos}(2\pi \phantom{\rule{0.25em}{0ex}}\text{ft})\text{,}\)
where \(f=1/T\) is the frequency of the wave. Adding two waves that have different frequencies but identical amplitudes produces a resultant
\(x={x}_{1}+{x}_{2}\text{.}\)
More specifically,
\(x=X\phantom{\rule{0.25em}{0ex}}\text{cos}(2\pi \phantom{\rule{0.25em}{0ex}}{f}_{1}t)+X\phantom{\rule{0.25em}{0ex}}\text{cos}(2\pi \phantom{\rule{0.25em}{0ex}}{f}_{2}t)\text{.}\)
Using a trigonometric identity, it can be shown that
\(x=2X\phantom{\rule{0.25em}{0ex}}\text{cos}(\pi \phantom{\rule{0.25em}{0ex}}{f}_{\text{B}}t)\text{cos}(2\pi \phantom{\rule{0.25em}{0ex}}{f}_{\text{ave}}t)\text{,}\)
where
\({f}_{B}=\mid {f}_{1}-{f}_{2}\mid \)
is the beat frequency, and \({f}_{\text{ave}}\) is the average of \({f}_{1}\) and \({f}_{2}\). These results mean that the resultant wave has twice the amplitude and the average frequency of the two superimposed waves, but it also fluctuates in overall amplitude at the beat frequency \({f}_{\text{B}}\). The first cosine term in the expression effectively causes the amplitude to go up and down. The second cosine term is the wave with frequency \({f}_{\text{ave}}\). This result is valid for all types of waves. However, if it is a sound wave, providing the two frequencies are similar, then what we hear is an average frequency that gets louder and softer (or warbles) at the beat frequency.
Making Career Connections
Piano tuners use beats routinely in their work. When comparing a note with a tuning fork, they listen for beats and adjust the string until the beats go away (to zero frequency). For example, if the tuning fork has a \(\text{256}\phantom{\rule{0.25em}{0ex}}\text{Hz}\) frequency and two beats per second are heard, then the other frequency is either \(\text{254}\) or \(\text{258}\phantom{\rule{0.25em}{0ex}}\text{Hz}\). Most keys hit multiple strings, and these strings are actually adjusted until they have nearly the same frequency and give a slow beat for richness. Twelve-string guitars and mandolins are also tuned using beats.
While beats may sometimes be annoying in audible sounds, we will find that beats have many applications. Observing beats is a very useful way to compare similar frequencies. There are applications of beats as apparently disparate as in ultrasonic imaging and radar speed traps.
Check Your Understanding
Imagine you are holding one end of a jump rope, and your friend holds the other. If your friend holds her end still, you can move your end up and down, creating a transverse wave. If your friend then begins to move her end up and down, generating a wave in the opposite direction, what resultant wave forms would you expect to see in the jump rope?
Solution
The rope would alternate between having waves with amplitudes two times the original amplitude and reaching equilibrium with no amplitude at all. The wavelengths will result in both constructive and destructive interference
Check Your Understanding
Define nodes and antinodes.
Solution
Nodes are areas of wave interference where there is no motion. Antinodes are areas of wave interference where the motion is at its maximum point.
Check Your Understanding
You hook up a stereo system. When you test the system, you notice that in one corner of the room, the sounds seem dull. In another area, the sounds seem excessively loud. Describe how the sound moving about the room could result in these effects.
Solution
With multiple speakers putting out sounds into the room, and these sounds bouncing off walls, there is bound to be some wave interference. In the dull areas, the interference is probably mostly destructive. In the louder areas, the interference is probably mostly constructive.
PhET Explorations: Wave Interference
Make waves with a dripping faucet, audio speaker, or laser! Add a second source or a pair of slits to create an interference pattern.
This lesson is part of:
Oscillatory Motion and Waves