Infer standing-wave modes from displacement and pressure boundaries

Establish the model and evidence

Standing waves can form from oppositely traveling waves in a constrained region. Their nodes remain at fixed positions, while antinodes are locations of maximum oscillation amplitude. An antinode is not permanently at maximum displacement: it passes through equilibrium during each cycle. The allowed wavelengths follow the end conditions, not an arbitrary list of frequencies chosen by the observer.

A string fixed at both ends has displacement nodes at each end. For length L, an integer number of half-wavelengths fits: λn=2L/n and fn=nv/(2L). An ideal open-open air column also supports all integer harmonics, but its ends are displacement antinodes and pressure nodes. The same frequency sequence does not mean the same spatial quantity has nodes at the same endpoints.

A closed-open air column has a displacement node at the closed end and an antinode at the open end. The pressure conditions are complementary. Its allowed wavelengths are 4L/(2k−1), where k counts allowed modes. Thus the second allowed mode is the third harmonic, not the second harmonic. A model that simply assigns all integer multiples of the fundamental would violate these mixed end conditions.

The plots show normalized spatial factors for displacement and, separately, pressure. They are not an assertion that both quantities reach their maxima at the same instant. Real pipes have end corrections, losses and geometry-dependent shifts, all omitted here. Inferring a length or wave speed from resonances therefore requires checking the boundary type and whether consecutive observed peaks are actually consecutive allowed modes.

Work a reference case

For an ideal 1.0 m closed-open air column with v=340 m/s, the fundamental is 85 Hz. Allowed-mode index 2 is harmonic 3, with λ=4/3 m and f=255 Hz. A 1.0 m open-open column instead has fundamental 170 Hz and its second mode is 340 Hz.

Predict before investigating

Does an open end of an air column impose a pressure node or a displacement node?

Compare all three boundary choices at mode index 2. Read displacement endpoints and, for air, pressure endpoints. Then compare indices 1–4 for the closed-open case and infer the spacing between allowed resonances.

Transfer to changed conditions

For a 0.50 m closed-open air column at 340 m/s, calculate the fundamental and next allowed frequency. Identify the pressure condition at the open end.

Compare your reasoning

The fundamental is 170 Hz and the next allowed frequency is 510 Hz. The open end is a pressure node and a displacement antinode in the idealized model.

Reflect and qualify the evidence

What evidence would distinguish a missing measured resonance from a genuinely forbidden harmonic?

Transfer a mixed end condition to a string

A loose or ideal free string end is a displacement antinode, unlike a fixed end’s displacement node. Predict and check: a virtual string of length 0.80 m and speed 40 m/s is fixed at one end and free at the other. Its fundamental wavelength is 4L=3.20 m and frequency 12.5 Hz. The next allowed mode is the third harmonic at 37.5 Hz, not a 25 Hz second harmonic. Sketch one quarter-wave and then three quarter-waves, preserving the unlike endpoints. If both ends are fixed instead, the fundamental is 25 Hz and all positive integer harmonics are allowed. This is a boundary comparison, not a request to construct an unstable loose-string apparatus. (Current clause 14.6.B.)

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.

Clause-level source/action reconciliation checked 31 October 2026 against the linked current framework and Fall 2026 clarification. Added packets are original virtual comparisons; no new practical or learner-completion claim.