Resolve wave depth, propagation and refraction

1. Define the process and evidence

A wave crest travels at phase speed, while the envelope and energy of a small wave group travel at group speed. Neither is automatically the speed at which all water particles move toward shore. Water motion in linear surface waves is largely oscillatory, with geometry depending on depth and wavelength.

2. Connect mechanism and boundary

The original small-amplitude gravity-wave model uses ω=2π/T and solves ω²=gk tanh(kh), where k is wavenumber and h depth. Phase speed is ω/k and wavelength 2π/k. Group speed is half the phase speed multiplied by [1+2kh/sinh(2kh)]. The bounded numerical solver brackets k between 0 and 1 rad/m for every supplied period/depth pair.

In deep water, phase speed approaches gT/(2π) and group speed approaches half of phase speed. In shallow water both approach √(gh). At a fixed period, entering shallower water reduces phase speed and wavelength. With parallel depth contours and no current, sinθ/c is conserved, so a ray initially 30° from the shore normal bends toward that normal as it slows.

Wind, fetch and duration influence wave generation, while refraction can redistribute energy along irregular coasts. This linear model excludes breaking, currents, storm surge, diffraction and harbor geometry. It is a method comparison, not a forecast or a safe-navigation tool. The conventional amplitude is half wave height, and no source image or ambiguous height convention is copied here.

3. Reconstruct the worked packet

For a 10 s wave in 20 m depth, the solver gives k≈0.051826 rad/m, wavelength≈121.237 m and phase speed≈12.1237 m/s. Group speed is about 9.2745 m/s. The deep-water comparison speed is 15.6131 m/s, so refraction from 30° gives an angle smaller than 30° to the shore normal.

4. Predict and investigate

Predict first: Do crest speed and group speed become identical in every water depth?

Compare the three periods at fixed depth, then vary depth at fixed period. Substitute solved k into the dispersion equation and retain phase/group quantities separately. State why this does not predict breaking height or a safe route.

5. Test a changed case

Apply the evidence to a changed case

In the shallow-water limit, how do phase and group speeds compare, and what determines their common leading value?

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

Both approach √(gh), controlled primarily by depth in that limit. The finite-depth slider still uses the full dispersion relation; do not replace it by the limit without checking kh is small.

6. Retain a defensible process notebook

Preserve first predictions, complete source/units, geometry, flux and storage boundaries, calculations and competing interpretations. Only the explicitly labeled USGS discharge archive is actual measured-derived source data; all other numerical packets and diagrams are invented. Public practice does not certify independent reasoning, actual field competence or a safe environmental decision.

Mechanism or dataset reference, checked 30 October 2026. Original prose, calculations, diagrams and questions; source observations retain agency and provenance.