Explain the phenomenon of beats in sound waves and how the human ear perceives them, including an example.

Beats are produced when two sound waves of slightly different frequencies interfere. This interference causes a periodic variation in the amplitude of the resultant wave, which is perceived as a periodic rise and fall in loudness. The beat frequency ($f_{beat}$) is the absolute difference between the frequencies of the two interfering waves:

$$f_{beat} = |f_1 - f_2|$$

where $f_1$ and $f_2$ are the frequencies of the two sound waves.

The human ear can only distinguish individual beats when the rate of loudness variation is slow enough.

• When the frequency difference ($f_{beat}$) is less than approximately 10 Hz, the loudness changes at a slow pace, allowing our ears to perceive a distinct, pulsing sound.
• When the frequency difference ($f_{beat}$) is greater than 10 Hz, the loudness changes too rapidly. The individual pulses of loudness merge into a continuous sound, and our ears cannot distinguish them as separate beats. Instead, we perceive a rough or dissonant sound, or sometimes two distinct tones if the frequency difference is very large.

Consider two tuning forks:

• If one tuning fork has a frequency of $440 \text{ Hz}$ and another has $442 \text{ Hz}$, the beat frequency is $f_{beat} = |440 \text{ Hz} - 442 \text{ Hz}| = 2 \text{ Hz}$. Our ears can easily perceive these 2 beats per second as a clear pulsing sound.
• If one tuning fork has a frequency of $440 \text{ Hz}$ and another has $455 \text{ Hz}$, the beat frequency is $f_{beat} = |440 \text{ Hz} - 455 \text{ Hz}| = 15 \text{ Hz}$. In this case, the loudness changes 15 times per second, which is too fast for the human ear to distinguish as individual beats. We would hear a continuous, possibly rough, sound rather than distinct pulses.

The final answer is $\boxed{\text{It is difficult to recognize beats when the frequency difference is greater than 10 Hz because the loudness changes too rapidly for the human ear to distinguish individual pulses.}}$