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.
Description
Finite-depth waves have phase and group speeds governed by dispersion.
Finite-depth waves have phase and group speeds governed by dispersion. The wave number must satisfy both period and water depth.
Solve finite-depth linear dispersion omega²=g k tanh(kh) by bounded dimensionless bisection, returning phase/group speeds without deep/shallow aliases.
When to use Wave Dispersion Relation Root Solver
Use this tool to solve finite-depth linear dispersion omega²=g k tanh(kh) by bounded dimensionless bisection, returning phase/group speeds without deep/shallow aliases.
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 Wave Dispersion Relation Root Solver 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 periodSeconds, waterDepthM, gravityMPerSecondSquared using the field table and calculation rules, then run the tool.
- Read waveNumberPerM, wavelengthM, phaseSpeedMPerSecond, groupSpeedMPerSecond. 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 |
|---|---|---|---|
| periodSeconds | Period seconds | number | Required |
| waterDepthM | Water depth m | number | Required |
| gravityMPerSecondSquared | Gravity m per second squared | number | Required |
Use positive period, depth and gravity. Bracket the dimensionless root kdepth and perform 100 bisection steps. Return wave number, 2π/k wavelength, phase speed and finite-depth group speed.1
angular frequency² = gravity × k × tanh(kdepth) Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.
| Output field | Meaning | Value type |
|---|---|---|
| waveNumberPerM | Wave number per m | number |
| wavelengthM | Wavelength m | number |
| phaseSpeedMPerSecond | Phase speed m per second | number |
| groupSpeedMPerSecond | Group speed m per second | number |
{
"input": {
"periodSeconds": 6.283185307179586,
"waterDepthM": 1,
"gravityMPerSecondSquared": 1.3130352854993315
},
"output": {
"waveNumberPerM": 1,
"wavelengthM": 6.283185307179586,
"phaseSpeedMPerSecond": 1,
"groupSpeedMPerSecond": 0.7757205647717832
}
}Limitations and assumptions
Linear waves and uniform supplied depth are assumed.
Numbers and computed results must be finite; overflow produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.
Alternative or Complementary methods
Use Wave Orbital Velocity At Bottom: Calculate finite-depth Airy bottom orbital velocity amplitude using solved dispersion and an overflow-safe inverse-sinh form; linear-wave assumptions exclude breaking and strong nonlinearity. For a complementary task, use Shoaling Coefficient H H0: Transform wave height by supplied group speeds and retained energy flux on constant-width rays; no refraction or breaking correction is silently included.
References
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Wave Dispersion Relation Root Solver — tool.io
Similar or alternative tools
- Wave Orbital Velocity At Bottom
Calculate finite-depth Airy bottom orbital velocity amplitude using solved dispersion and an overflow-safe inverse-sinh form; linear-wave assumptions exclude breaking and strong nonlinearity.
- Shoaling Coefficient H H0
Transform wave height by supplied group speeds and retained energy flux on constant-width rays; no refraction or breaking correction is silently included.