Significant Height Spectral Moment

Integrate a supplied one-sided elevation PSD by trapezoidal spectral moments, returning Hm0 and Tm02; missing frequency bands are not extrapolated and buoy-heave aliases excluded.

Description

A one-sided elevation spectrum describes wave variance by frequency.

A one-sided elevation spectrum describes wave variance by frequency. Its zeroth and second moments estimate height and zero-crossing period.

Integrate a supplied one-sided elevation PSD by trapezoidal spectral moments, returning Hm0 and Tm02; missing frequency bands are not extrapolated and buoy-heave aliases excluded.

When to use Significant Height Spectral Moment

Use this tool to integrate a supplied one-sided elevation PSD by trapezoidal spectral moments, returning Hm0 and Tm02; missing frequency bands are not extrapolated and buoy-heave aliases excluded.

Gather the water, wave, current or profile observations and any supplied calibration constants. Confirm units and the stated coordinate and pressure conventions.

How to use Significant Height Spectral Moment and understand its results

Gather the water, wave, current or profile observations and any supplied calibration constants. Confirm units and the stated coordinate and pressure conventions.
Enter frequenciesHz, elevationSpectralDensitiesM2PerHz using the field table and calculation rules, then run the tool.
Read zerothMomentM2, secondMomentM2Hz2, significantWaveHeightM, zeroCrossingPeriodSeconds. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
frequenciesHz	Frequencies hz	list of number	Required
elevationSpectralDensitiesM2PerHz	Elevation spectral densities m2 per hz	list of number	Required

Align increasing positive Hz with nonnegative m²/Hz values. Integrate the spectrum and frequency-squared spectrum by trapezoids, requiring positive moments. Return moments, height and seconds.^[1]

Hm0 = 4sqrt(m0); Tm02 = sqrt(m0/m2)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
zerothMomentM2	Zeroth moment m2	number
secondMomentM2Hz2	Second moment m2 hz2	number
significantWaveHeightM	Significant wave height m	number
zeroCrossingPeriodSeconds	Zero crossing period seconds	number

json

{
  "input": {
    "frequenciesHz": [
      0.1,
      0.2,
      0.3
    ],
    "elevationSpectralDensitiesM2PerHz": [
      1,
      1,
      1
    ]
  },
  "output": {
    "zerothMomentM2": 0.19999999999999998,
    "secondMomentM2Hz2": 0.009,
    "significantWaveHeightM": 1.7888543819998317,
    "zeroCrossingPeriodSeconds": 4.714045207910317
  }
}

Limitations and assumptions

Unobserved frequency bands are not extrapolated.

Numbers and computed results must be finite; overflow produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Pierson Moskowitz Significant Height: Integrate a caller-parameter Pierson–Moskowitz omega^-5 spectrum analytically over full and finite frequency bands; reference wind-height convention is caller calibrated, not a measured buoy-spectrum alias. For a complementary task, use Wave Dispersion Relation Root Solver: Solve finite-depth linear dispersion omega²=g k tanh(kh) by bounded dimensionless bisection, returning phase/group speeds without deep/shallow aliases.

References

Significant Height Spectral Moment — tool.io

Similar or alternative tools

Pierson Moskowitz Significant Height
Integrate a caller-parameter Pierson–Moskowitz omega^-5 spectrum analytically over full and finite frequency bands; reference wind-height convention is caller calibrated, not a measured buoy-spectrum alias.
Wave Dispersion Relation Root Solver
Solve finite-depth linear dispersion omega²=g k tanh(kh) by bounded dimensionless bisection, returning phase/group speeds without deep/shallow aliases.