Explain inversion without confusing amplitude and transmitted energy

Establish the model and evidence

A pulse approaching a join can produce both reflected and transmitted disturbances. For two ideal strings under the same tension, the denser string has lower wave speed. Reflection from the slower side is inverted; reflection from a faster side is not. Equal strings have no reflected pulse in this idealization. Fixed-end and free-end reflection are useful limiting cases, not evidence that every boundary has the same sign.

The source frequency remains continuous across a stationary boundary. Because speeds differ, wavelengths differ in the same ratio as speed. A pulse of a given duration is spatially narrower in the slower string. In the plot, pulses are centered separately to compare shape and sign: they are not three contributions occupying one place at the same time, so adding those displayed profiles would misinterpret the diagram.

The supplied quantitative coefficients are optional enrichment beyond the core qualitative boundary reasoning. With speeds v1 and v2 and equal tension, reflected displacement coefficient r=(v2−v1)/(v2+v1), while transmitted coefficient t=1+r. These follow displacement continuity and the string boundary constraints. The signs refer to displacement relative to the incident pulse, not to negative energy.

Power fractions are r² and (v1/v2)t² in this equal-tension model. They add to one, even when the transmitted displacement amplitude exceeds the incoming amplitude. A different medium converts displacement amplitude to energy flux differently. Therefore a tall transmitted pulse alone is not an energy-creation claim, and squaring both displacement coefficients without the speed factor would give a wrong ledger. No physical string tension or apparatus is requested.

Work a reference case

For density ratio μ2/μ1=4 and v1=10 m/s, v2=5 m/s. The optional coefficients are r=−1/3 and t=2/3; reflected power is 1/9 and transmitted power is 8/9. At 4 Hz, wavelengths are 2.5 m and 1.25 m, with unchanged frequency.

Predict before investigating

Can a transmitted displacement amplitude exceed the incident amplitude while total power remains conserved?

Compare density-ratio choices 0, 2 and 4 at 4 Hz. Retain sign, width and both power fractions. Then double frequency at one boundary and identify which coefficients stay fixed and which wavelengths change.

Transfer to changed conditions

A stationary equal-tension join has wave speeds 12 and 6 m/s with source frequency 3 Hz. Predict reflection sign and both wavelengths before using any coefficient formula.

Compare your reasoning

The reflected pulse is inverted. Incident wavelength is 4 m and transmitted wavelength is 2 m; frequency remains 3 Hz. The optional coefficients would again be −1/3 and 2/3.

Reflect and qualify the evidence

Why must an energy ledger use the boundary’s conversion factors rather than just add squared plotted heights?

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.