Add signed disturbances and distinguish beats from cancellation

Establish the model and evidence

In a linear wave model, disturbances at the same position and time add algebraically. Two pulses pass through one another rather than collide like rigid objects. For example, an upward displacement of 3 mm overlapping a downward displacement of 2 mm gives +1 mm at that instant. After separation, the original pulses continue under the assumed linear, nondissipative model; the overlap is not evidence they bounced.

The interactive trace uses two pressure contributions at one fixed position. Each has amplitude 1 Pa, but their phases and frequencies can differ. The total trace is their signed sum, not the sum of positive peak heights at every instant. Equal-frequency waves with zero phase difference reinforce; a 180° difference gives cancellation at that observation point. An intermediate phase gives another fixed resultant amplitude rather than a periodic beat envelope.

When frequencies differ slightly, their relative phase drifts. They repeatedly reinforce and oppose, producing beats at the absolute frequency difference. The slow envelope describes amplitude variation of the fast oscillation, not a third independent source transporting all energy at the beat frequency. The phase setting shifts when maxima occur but does not change the difference in frequencies.

A local cancellation does not establish that total energy of the complete sources and field vanished. Spatial redistribution, source work and the broader wave configuration matter. This small-amplitude pressure model ignores nonlinear acoustics and makes no sound output. A comparison of two plotted components should be made at equal times; selecting one crest from each at different times would not test superposition.

Work a reference case

For 20 Hz and 24 Hz components, beat rate is 4 Hz and the beat period is 0.25 s. At equal 20 Hz with a 180° relative phase, the signed sum at the modeled point is zero throughout, not a sequence of 20 beats per second.

Predict before investigating

If two equal-frequency contributions are opposite in phase, will their relative phase gradually drift?

Capture frequency differences 0, 2 and 4 Hz at zero phase, then compare 0 and 180° at equal frequency. Check several same-time coordinate rows by adding components yourself. Explain which changes alter beat spacing and which alter phase timing.

Transfer to changed conditions

Two hypothetical tones have frequencies 440 and 446 Hz. Find the beat rate. Separately, two overlapping pulses have displacements +5 mm and −8 mm; find the resulting displacement.

Compare your reasoning

Beat rate is 6 Hz. The instantaneous pulse displacement is −3 mm; that signed value does not imply a negative energy or permanent loss of either pulse.

Reflect and qualify the evidence

Why is a small or zero local sum not sufficient to claim the waves destroyed energy everywhere?

Sources and boundaries

Original text, calculations and diagrams; checked 30 October 2026 against the current AP Physics 2 course framework and primary textbook verification. Controls generate virtual evidence, not actual measurements or independent practical competence. No lasers, sunlight focusing, loud audio, traffic observations or chemical-film preparation are requested. Public feedback is practice; retain your derivation and critique in the course evidence workspace.