What are waxing and waning in the phenomenon of beats? Show that waxing and waning occur periodically.

In the context of wave phenomena, such as sound waves, "waxing" and "waning" refer to the changes in the intensity or loudness of a sound when two slightly different frequencies of waves interfere constructively and destructively. This phenomenon is known as "beats." Let's explore what waxing and waning mean in the context of beats and how they occur periodically.

Waxing and Waning in Beats:

Waxing: When two waves with slightly different frequencies overlap, there are moments when their peaks and troughs align, causing constructive interference. This results in an increase in the overall amplitude (loudness) of the combined wave. This increase in amplitude is known as "waxing" in the context of beats.

Waning: Conversely, when the peaks of one wave align with the troughs of the other wave, they interfere destructively, causing a decrease in the overall amplitude of the combined wave. This decrease in amplitude is known as "waning" in the context of beats.

Periodicity of Waxing and Waning:

Waxing and waning in beats occur periodically because they are directly related to the difference in frequency between the two interfering waves. The frequency of the beat is equal to the absolute difference between the frequencies of the two waves involved. Mathematically, if f1 and f2 are the frequencies of the two waves, the beat frequency (f_beat) is given by:

f_beat = |f1 - f2|

To illustrate the periodicity of waxing and waning, consider an example:

Let's say we have two tuning forks, one with a frequency of 440 Hz (A4) and another with a frequency of 442 Hz. The beat frequency would be:
f_beat = |440 Hz - 442 Hz| = 2 Hz

This means that the intensity (loudness) of the combined sound will wax and wane at a rate of 2 cycles per second.

As time progresses, you will hear the sound go from loud to soft (waxing and waning) with a regular pattern. The period of this waxing and waning is the inverse of the beat frequency:
Period = 1 / f_beat = 1 / 2 Hz = 0.5 seconds

So, in this example, the waxing and waning of the sound will repeat every 0.5 seconds.

In summary, waxing and waning in the phenomenon of beats occur periodically, and the period of this waxing and waning is inversely proportional to the beat frequency, which is determined by the difference in frequencies between the interfering waves. This periodic variation in loudness is a key characteristic of beat phenomena and is often used in tuning musical instruments.