Standing Wave Harmonics Calculator (strings & pipes) mechanism

Model scope

The core frequency formulas are correct for the stated ideal boundary conditions. The instrument needs an explicit wavelength formula, a defined harmonic-table range, validation rules for L, v, and n, and documented speed-of-sound table conditions before release.

• The stated frequency formulas are standard: f_n = nv/(2L) for a string with both ends fixed and an open pipe, and f_n = (2n−1)v/(4L) for a pipe closed at one end.
• The claimed wavelength output has no specified formula; it should use λ = v/f_n, giving λ = 2L/n for strings/open pipes and λ = 4L/(2n−1) for closed pipes.
• The harmonic-series table has no specified range or maximum harmonic number, so its output is ambiguous.
• For a closed pipe, n must be the positive mode index n = 1,2,3,...; these correspond to the first, third, fifth, ... harmonics, not all integer harmonics.
• The speed-of-sound table is underspecified unless its medium, temperature, and environmental assumptions are defined. No frequency formula itself is incorrect.

Units and conventions

• L must be positive and expressed in a length unit compatible with v; v must be expressed in the corresponding length unit per second.
• n is dimensionless and must be a positive integer.
• For strings and open pipes, the stated formula assumes ideal 1D standing waves with nodes at both string ends or antinodes at both pipe ends.
• For closed pipes, the formula assumes one pressure node and one pressure antinode, with only odd modes present.
• Wavelength is in the same length unit as L because λ = v/f_n.
• The supplied speed-of-sound value must correspond to the selected medium and conditions; air speed varies with temperature and other environmental factors.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "boundary_type": "string_both_ends_fixed",
    "L": 0.65,
    "L_unit": "m",
    "v": 130,
    "v_unit": "m/s",
    "n": 3,
    "harmonic_table_through_n": 3
  },
  "expected_outputs": {
    "f_n_Hz": 300,
    "wavelength_m": 0.4333333333,
    "harmonic_series": [
      {
        "n": 1,
        "f_Hz": 100,
        "wavelength_m": 1.3
      },
      {
        "n": 2,
        "f_Hz": 200,
        "wavelength_m": 0.65
      },
      {
        "n": 3,
        "f_Hz": 300,
        "wavelength_m": 0.4333333333
      }
    ]
  },
  "working": "String formula: f_3 = (3×130)/(2×0.65) = 390/1.30 = 300 Hz. Wavelength: λ_3 = v/f_3 = 130/300 = 0.4333333333 m, equivalently 2L/n = (2×0.65)/3 = 1.30/3 = 0.4333333333 m. Table: n=1 gives 130/1.30 = 100 Hz; n=2 gives 260/1.30 = 200 Hz; n=3 gives 390/1.30 = 300 Hz."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

A zero-length input produces division by zero and undefined frequency. The instrument must reject L=0; otherwise it can emit an invalid result or fail. Signed wave speed also exposes a convention hazard because literal substitution yields a negative frequency.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json