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Beats

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[Note: The Java Applet for this animation can no longer be run in a web browser. Clicking this link, however, will download to your computer a Java Web Start file named BeatsLaunch.jnlp. This file can launch the animation in a separate window on your computer. Downloading and executing the file may be blocked by local Java permissions, but persevere! You may finally have to launch the file by right-clicking (control-clicking) it and choosing Open, instead of simply double-clicking it.]

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Consider two processes that vary sinusoidally at slightly different frequencies.  If at any time they are vibrating in step with each other they do not remain that way, since the lower-frequency process will gradually fall behind the higher-frequency one.  Eventually, however, the lower-frequency process will fall behind the other by a whole cycle and then they will be vibrating in step again.  This sequence repeats itself, and the two processes keep changing from in step, to out of step, to back in step, then out of step, etc.

If there is some reason to consider the sum of the two processes, the sum is twice as large as either process alone when they are in step, and the sum is zero when they are exactly out of step.  Thus the amplitude of the sum varies regularly and slowly from zero to a large value and back again.

This behavior is described succinctly by the trigonometric identity:

Here the difference between two sine functions is evaluated, since that is the case pictured in the animation.  The right-hand side is the product of two terms, a cosine term that oscillates with the average frequency of the two component sine functions, multiplied by a sine term that varies at a low frequency if the two component frequencies are nearly equal.  Thus the sine factor may be viewed as a slowly changing amplitude for the cosine factor.  The sine factor is large in absolute value twice in each of its cycles, and so the frequency with which the sum has large amplitude is equal to the difference between the two component frequencies.

If the two processes are sound waves of slightly different frequencies, the amplitude of the net wave varies slowly.  The resulting pulsing changes in loudness are called “beats.” 

Operation

1. At the bottom of the window are several controls for the display.  On the left is a button labeled “Go/Stop.”  Clicking the button at any time stops or restarts the motion.  Next to that is a button labeled “Show/Hide Sum.”   Clicking that turns on or off a display of the sum of the two component sine waves.  The button labeled “Clear” restarts the graphing at any time.

The remaining control at the bottom of the window is only effective when the sum is being displayed.  The slider may be used to move a vertical white line anywhere along the graphs, and the values read from the graphs at the location of the line are displayed at the right end of the axes.  Thus one can check at any point that the top graph is indeed the sum of the bottom two graphs.

2. At the top of the window is a menu bar that affects the display.  Available menu choices and their effects are the following:

Can you verify that the frequency of the beats is the difference between the two component frequencies, and the frequency of the sum is the average of the two component frequencies?