Tsunami Wave Shoaling & Runup

Wave & Shelf Parameters

Offshore wave height H₀ 2.0 m

Wave period T 600 s

Shelf depth h₀ 60 m

Beach slope β 3.0°

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Non-breaking shoaling

Wave height at breaking H_b

– m

Breaking depth h_b

– m

Shallow-water speed c=√(gh)

– m/s

Max vertical runup R

– m

Shoaling coeff. K_s at break

–

Iribarren-type ratio H₀/h₀

–

How it works

A tsunami behaves as a shallow-water wave (wavelength ≫ depth) the entire way from the deep ocean to the coast, so its local speed depends only on local depth h:

c(x) = √(g·h(x))          (shallow-water celerity)

As the wave crosses a shoaling shelf its energy flux is conserved, which forces the height to grow as the group speed drops — Green's Law:

H(x) = H₀ · (h₀ / h(x))^(1/4)

This blow-up stops once the wave becomes too steep to stay stable. It breaks when the height-to-depth ratio reaches the breaker index γ_b ≈ 0.78:

H(x) = γ_b · h(x),   solved with Green's Law gives
h_b = ( H₀ · h₀^(1/4) / γ_b )^0.8

Past that point the crest collapses into a turbulent bore that keeps thinning as it crosses the surf zone, modeled here as H ≈ γ_b·h(x). The final vertical runup on the beach face is estimated with the classic Synolakis (1987) solitary-wave runup law for a plane beach of slope β:

R / h₀ = 2.831 · √(cot β) · (H₀ / h₀)^(5/4)

H₀, T — the incoming wave's height and period out on the shelf edge, before any shoaling.
Shelf depth h₀ — water depth where the wave begins climbing the slope toward shore.
Beach slope β — steeper beaches (larger β) shorten the surf zone but also shorten the runup for the same offshore wave; gentler beaches let the same wave push much farther inland.

Real-world relevance: this is exactly why tsunamis are barely noticeable in open ocean (H₀ often under a metre) yet arrive onshore as multi-metre bores — the identical physics used in coastal tsunami-hazard and inundation mapping.

Sea surface Breaker / foam Seafloor & beach

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