Why does the tsunami slow down and grow taller as it reaches the coast?

The wave speed is c = √(gH). As depth H decreases near shore, the speed drops. The wave's energy flux must be approximately conserved. Since the speed decreases, the wave amplitude must increase to maintain the energy transport rate, leading to the dramatic height increase called shoaling. This is analogous to a line of runners slowing down and bunching up.

Is this simulator realistic for all tsunamis?

It captures the essential linear physics of propagation and shoaling for small-amplitude tsunamis in the open ocean. However, it simplifies by being one-dimensional, non-dispersive, and linear. Real tsunamis can be influenced by 2D/3D bathymetry, nonlinear effects (which cause steepening and breaking very near shore), dispersion (which spreads out very long waves), and coastal run-up, which are not modeled here.

What does the 'Gaussian uplift impulse' represent?

It models a sudden, localized uplift of the seafloor, a common idealization of an earthquake source. The Gaussian shape is a smooth, mathematically convenient function that approximates a displaced volume of water. The simulator uses this initial condition for the water surface η(x,0) and then calculates how this disturbance evolves according to the wave equations.

Why are tsunamis considered 'shallow water' waves even in the deep ocean?

A wave is classified as a 'shallow water' wave when the water depth H is much less than its wavelength λ (typically H < λ/20). Tsunamis have wavelengths of hundreds of kilometers, while the deepest ocean is only about 10 km deep. Therefore, even in the deep ocean, the condition H << λ is satisfied, so the shallow water wave approximation and the formula c = √(gH) apply accurately.

Tsunami & Shallow Water (1D)

Linear η,u on H(x): c = √(gH) drops over shelf; Gaussian uplift impulse.

About this model

Tsunamis are long-wavelength gravity waves generated by sudden seafloor displacement, such as from earthquakes or landslides. This simulator models their fundamental one-dimensional propagation across a varying ocean depth, H(x). The core physics is governed by the linear shallow water wave equations, which are a simplification of the Navier-Stokes equations valid when the wavelength is much greater than the water depth. These equations describe the evolution of the water surface displacement, η(x,t), and the depth-averaged horizontal fluid velocity, u(x,t). A critical result from these equations is that the wave speed, or phase speed, is not constant but depends solely on the local depth: c = √(gH), where g is gravitational acceleration. As a tsunami travels from the deep ocean (large H, high c) onto a continental shelf (smaller H, lower c), it slows down. Conservation of energy and wave action then lead to a dramatic increase in wave amplitude, a process known as shoaling. The simulator initiates the wave with a Gaussian-shaped uplift impulse, representing a simplified seismic source. Key simplifications include the 1D geometry, the linearization of the equations (neglecting nonlinear steepening effects), and the omission of dispersion, friction, and wave breaking. By interacting with this model, students can directly observe the relationship between depth and wave speed, the shoaling process, partial reflection at depth changes, and how the initial seafloor deformation evolves into a propagating wave train.

Who it's for: Undergraduate students in geophysics, oceanography, or fluid dynamics courses studying wave dynamics and tsunami physics. It is also valuable for advanced high school physics students exploring real-world applications of wave mechanics.

Key terms

Shallow Water Wave Equations
Phase Speed
Shoaling
Wave Amplification
Continental Shelf
Gravity Wave
Tsunami Generation
Linear Wave Theory

How it works

Shallow-water intuition: a seabed uplift (Gaussian bump) launches a disturbance; phase speed √(gH) shrinks over the continental shelf, so energy crowds into a narrower, slower wave — the usual cartoon of tsunami approach.

Key equations

η_t + (Hu)_x = 0, u_t + gη_x = 0 (linear, depth H(x))

Frequently asked questions

Why does the tsunami slow down and grow taller as it reaches the coast?: The wave speed is c = √(gH). As depth H decreases near shore, the speed drops. The wave's energy flux must be approximately conserved. Since the speed decreases, the wave amplitude must increase to maintain the energy transport rate, leading to the dramatic height increase called shoaling. This is analogous to a line of runners slowing down and bunching up.
Is this simulator realistic for all tsunamis?: It captures the essential linear physics of propagation and shoaling for small-amplitude tsunamis in the open ocean. However, it simplifies by being one-dimensional, non-dispersive, and linear. Real tsunamis can be influenced by 2D/3D bathymetry, nonlinear effects (which cause steepening and breaking very near shore), dispersion (which spreads out very long waves), and coastal run-up, which are not modeled here.
What does the 'Gaussian uplift impulse' represent?: It models a sudden, localized uplift of the seafloor, a common idealization of an earthquake source. The Gaussian shape is a smooth, mathematically convenient function that approximates a displaced volume of water. The simulator uses this initial condition for the water surface η(x,0) and then calculates how this disturbance evolves according to the wave equations.
Why are tsunamis considered 'shallow water' waves even in the deep ocean?: A wave is classified as a 'shallow water' wave when the water depth H is much less than its wavelength λ (typically H < λ/20). Tsunamis have wavelengths of hundreds of kilometers, while the deepest ocean is only about 10 km deep. Therefore, even in the deep ocean, the condition H << λ is satisfied, so the shallow water wave approximation and the formula c = √(gH) apply accurately.

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Key equations

Frequently asked questions

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How it works

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Frequently asked questions