Wave shoaling¶

Shallowing water can slow a surface wave's group propagation and concentrate its energy into greater wave height along a sufficiently conserved path.

Version: v1 · 2026-10-07 · History
Domain-specific #: 14050
Domain group: Natural Sciences
Origin domain: Physics
Subdomains: Surface Gravity Waves, Coastal Oceanography → Physics

Core Idea¶

Wave shoaling is a wave-height gain caused by shallower water along a traveling surface-gravity wave's path. In the shoaling interval, group speed falls. If little energy is added or lost and the wave rays do not converge or spread much, energy is carried more densely and wave height can rise. This is a process acting on a wave, not every change that happens in shallow water.[^{ref-e990473603fb][}ref-b4de1f59c4ff]

For an ideal long wave, speed is approximately sqrt(gd) at water depth d; the one-dimensional, no-current Green-law comparison gives H/H₀=(d₀/d)^(1/4). That formula is a benchmark under its assumptions. Friction, currents, refraction and nonlinear changes can alter the actual height.[^ref-b4de1f59c4ff]

Scope of Application¶

Shoaling applies to surface-gravity waves crossing changing bathymetry, including laboratory wave trains over a ramp and modeled tsunami-like waves moving over an ocean slope. A tsunami can count as a shallow-water wave by its wavelength-to-depth ratio even in geographically deep water. The entry is deliberately narrower than the frozen source title “Waves and shallow water,” which also covers refraction, breaking and other wave behavior.[^{ref-a43b9750c17c][}ref-5a267371b1d0][^ref-b4de1f59c4ff]

Clarity¶

A taller shoreward wave does not by itself prove shoaling. Ask whether the height gain can be attributed to the depth-driven group-speed change, after accounting for ray focusing, wind, current and loss. Janssen also describes a shorter-period wave's initial action-density decrease before the later increase he names shoaling, so the entire trip from deep water need not be a monotonic gain.[^ref-e990473603fb]

Manages Complexity¶

Use four checks: identify the propagating wave; locate the depth change; establish the group-speed and energy or action-flux conditions; and determine whether the observed height response is attributable to that change. Treat friction, refraction and breaking as separate contributions or limits. The simple long-wave law is useful when its conditions hold, not a replacement for the full path budget.[^{ref-e990473603fb][}ref-b4de1f59c4ff]

Abstract Reasoning¶

When a wave approaches shallower water, calculate or estimate whether its group speed falls in that interval. Under a sufficiently conserved flux and nearly fixed ray width, a local height gain is expected. If height later falls, that can mean friction or breaking has become dominant without erasing an earlier local shoaling gain. For strongly changing waveforms, compare with a model rather than applying Green's quarter-power law unconditionally.[^{ref-e990473603fb][}ref-a43b9750c17c][^ref-5a267371b1d0]

Knowledge Transfer¶

The same carrier, depth transition, flux condition and height response organize both a laboratory ramp and a modeled long-period ocean wave. Their numerical gains are case-specific; the flume's roughly 3% change is not a general coastal prediction. The more portable parent is Wave, which can exist without bathymetric shoaling.[^{ref-a43b9750c17c][}ref-5a267371b1d0]

Example¶

In large-flume control experiments without artificial seagrass mats, Villanueva and colleagues report that a ramp raised measured wave height by about 3% at sensor USS1 relative to offshore USS6. Wall and bed friction then caused wave decay farther along the control path. Mapped back: the surface-wave train is the carrier, the ramp changes depth, the source attributes the local height gain to shoaling, and the later decay shows the loss boundary. The paper does not separately establish that every control wave contributing to this result was unbroken or measure the group-speed flux in this comparison.[^{ref-a43b9750c17c][}ref-e990473603fb]

Relationships to Other Abstractions¶

Current abstraction Wave shoaling Domain-specific

Parents (1) — more general patterns this builds on

Wave shoaling presupposes Wave Prime

Bathymetric height amplification presupposes a propagating wave whose group speed and height can change; waves also exist without shoaling.

Hierarchy path (1) — routes to 1 parentless root

Wave shoaling → Wave

Neighborhood in Abstraction Space¶

Wave shoaling sits in a sparse region of the domain-specific corpus (93^rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Refraction bends a wave's direction and can focus rays. Runup is a shoreline response, not an offshore shoaling measurement. Breaking and friction can remove energy and limit the simple gain law. A wave in shallow water at constant depth is not necessarily shoaling, and an earlier intermediate-depth action decrease is not the positive shoaling interval.[^{ref-e990473603fb][}ref-b4de1f59c4ff][^ref-6677bd6aaf34]

References¶

[^ref-e990473603fb]: Peter A. E. M. Janssen, The Interaction of Ocean Waves and Wind (2004), printed pp. 20, 26 and 40–41, Eqs. 2.26, 2.46 and 2.84 and the Shoaling and Refraction discussions (one-based PDF pp. 27, 33 and 47–48). The action-density turnover near kD≈1.5 is on printed p. 41. https://airsea.ucsd.edu/wp-content/uploads/sites/10/2019/09/Janssen_2004.pdf

[^ref-b4de1f59c4ff]: F. I. González et al. (Science Review Working Group), Scientific and Technical Issues in Tsunami Hazard Assessment of Nuclear Power Plant Sites (2007), NOAA Technical Memorandum OAR PMEL-136, §5.4.4, printed pp. 68–69 (one-based PDF pp. 74–75), Eqs. 5.5–5.6 and text preceding Table 5-3. The quarter-power relation is for the stated no-current long-wave comparison; the report also discusses current and very shallow nonlinear limits. https://www.pmel.noaa.gov/pubs/PDF/gonz3031/gonz3031.pdf

[^ref-a43b9750c17c]: R. Villanueva, M. Paul and T. Schlurmann, “Wave dynamics alteration by discontinuous flexible mats of artificial seagrass can support seagrass restoration efforts,” Scientific Reports 13 (2023), article 19418, DOI 10.1038/s41598-023-46612-z, Methods and Results → Wave decay, especially one-based PDF p. 9 on no-mat control ramp gain and later viscous loss. https://www.vliz.be/imisdocs/publications/ocrd/394234.pdf

[^ref-5a267371b1d0]: P. A. Madsen, D. R. Fuhrman and H. A. Schäffer, “On the solitary wave paradigm for tsunamis,” Journal of Geophysical Research: Oceans 113 (2008), DOI 10.1029/2008JC004932, §4.1 paragraph [51] and Fig. 5 for the transient Green-law limit; §4.2 paragraph [52] and Fig. 6 for the separate 13-minute periodic simulation; §4.3 paragraph [53] for solitary/cnoidal comparisons. https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2008JC004932

[^ref-6677bd6aaf34]: Robert L. Wiegel, “Experimental study of surface waves in shoaling water,” Eos, Transactions American Geophysical Union 31, no. 3 (1950), 377–385, DOI 10.1029/TR031i003p00377, original author abstract only, for sloping-beach profile comparison and qualified steep-slope/breaker limit. The accessible abstract does not quantify height gain. https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/TR031i003p00377