When two waves with nearly equal frequencies move through the same region, their superposition can make the combined wave alternate between larger and smaller amplitudes, producing the familiar effect called beats.

Amplitude Variations from Overlapping Traveling Waves

When two traveling waves occupy the same place at the same time, the medium responds to the sum of their displacements. If the waves have slightly different frequencies, that sum does not stay at one fixed amplitude. Instead, the amplitude of the resulting wave changes over time.

At some moments, the waves line up so their displacements reinforce one another, creating a larger amplitude. A short time later, they are less aligned, so the resulting amplitude becomes smaller. This repeating rise and fall is the key amplitude variation in this topic.

When these repeated large-small changes are noticeable, they are called beats.

The important idea is that the original waves still continue traveling. The amplitude variation appears because the waves are continually shifting in and out of phase with each other. The medium is not changing frequency on its own; the pattern comes from how the two waves combine at each moment.

Why the Amplitude Rises and Falls

A wave’s frequency tells how rapidly it oscillates. If two waves have almost the same frequency, one completes cycles slightly faster than the other. Because of that small difference, one wave gradually slips ahead in phase.

When the waves are nearly in phase, their displacements add to produce a large amplitude. When they become nearly out of phase, the displacements partially or completely cancel, so the amplitude becomes small. After more time passes, the phase relationship shifts again, and the amplitude grows again.

This cycle repeats:

• large amplitude when the waves reinforce

• smaller amplitude as the phase difference increases

• large amplitude again when the waves return to reinforcement

For sound waves, this changing amplitude is heard as changing loudness. The sound does not seem to pulse because the source turns on and off. Instead, the overlapping waves make the sound strength at a location fluctuate in a regular way.

Beat Frequency

The rate at which these amplitude changes occur is called the beat frequency. It tells how many beats happen each second. For AP Physics 2, the essential relationship is that the beat frequency equals the difference between the two original frequencies.

Only the size of the frequency difference matters, which is why the absolute value is used. It does not matter which of the two waves has the higher frequency.

Interpreting the Equation

If the two frequencies are very close together, $f_{beat}$ is small, so the amplitude changes slowly. If the frequencies are farther apart, $f_{beat}$ is larger, so the amplitude changes more rapidly.

If $f_1=f_2$, then $f_{beat}=0$. In that case, there is no beat pattern caused by a frequency difference. The waves combine into a steady result instead of a repeated rise-and-fall pattern.

The beat frequency is not the same as the frequency of either original wave. The original waves still oscillate at their own frequencies. The beat frequency measures only how quickly the amplitude pattern repeats.

What a Beat Pattern Looks Like

A beat pattern is often shown as a rapidly oscillating wave inside a slowly changing envelope.

A modulated sinusoid with its upper and lower envelope curves highlighted. In beat phenomena, the fast oscillation corresponds to the individual wave oscillations, while the envelope represents the slow amplitude variation that produces the perceived “pulsing” in loudness. Source

The fast oscillations come from the individual waves themselves, while the envelope shows the changing amplitude caused by their overlap.

This means a graph of displacement versus time for beats can show two different time scales:

• a fast back-and-forth motion from the original waves

• a slower modulation showing the amplitude variation

For sound, your ear mainly notices the slower modulation as pulses in loudness. For other wave situations, the same idea appears as repeated increases and decreases in the size of the disturbance.

What to Recognize on AP Physics 2 Questions

Questions on this subsubtopic usually focus on identifying when beats occur and interpreting what the beat frequency means physically.

You should be able to recognize that:

• beats require two traveling waves with slightly different frequencies

• beats are a form of amplitude variation caused by wave overlap

• the number of beats per second equals the difference between the two wave frequencies

• a smaller frequency difference gives a slower beat pattern

• a larger frequency difference gives a faster beat pattern

• identical frequencies do not produce a beat pattern from frequency difference

You should also connect the observation to the wave model: repeated changes in amplitude or loudness mean the two waves are alternately reinforcing and canceling as their relative phase changes.