Tsunamis are very long waves, often with wavelengths much larger than the ocean depth. In that limit, they are commonly modeled as shallow-water waves, even in the deep ocean. The key simplification is that the wave speed depends mainly on gravity and water depth, not directly on wavelength. The standard relation is
\[
v=\sqrt{gh},
\]
where \(g\) is gravitational acceleration and \(h\) is the local water depth. This formula explains why a tsunami can move extremely fast across the open ocean. For example, if the depth is \(4000\ \text{m}\), then
\[
v=\sqrt{(9.81)(4000)}\approx 198\ \text{m/s},
\]
which is roughly \(713\ \text{km/h}\). As the wave approaches shore and the water becomes shallow, the speed drops sharply. At a depth of \(10\ \text{m}\), the same formula gives
\[
v=\sqrt{(9.81)(10)}\approx 9.9\ \text{m/s}.
\]
So the wave slows dramatically near the coast. However, slowing down does not mean the tsunami becomes harmless. In fact, the surface elevation often grows as the wave enters shallower water. A simple shoaling model uses the scaling
\[
A \propto h^{-1/4},
\]
where \(A\) is the wave amplitude. This means the amplitude increases as the depth decreases. Comparing deep and shallow water gives
\[
\frac{A_{\text{coast}}}{A_{\text{deep}}}=\left(\frac{h_{\text{deep}}}{h_{\text{coast}}}\right)^{1/4}.
\]
For the sample values \(h_{\text{deep}}=4000\ \text{m}\) and \(h_{\text{coast}}=10\ \text{m}\),
\[
\left(\frac{4000}{10}\right)^{1/4}=400^{1/4}\approx 4.47.
\]
So the idealized amplification factor is about \(4.47\), not \(4\) exactly. If the offshore amplitude were \(1\ \text{m}\), the model would predict a coastal amplitude of about \(4.47\ \text{m}\). This is only a simplified shoaling estimate, but it captures the basic reason tsunamis can become much more destructive near shore.
A tsunami warning system also depends on travel-time estimation. If the wave travels a long offshore distance \(d\), a first approximation to the travel time is
\[
t=\frac{d}{v}.
\]
Using the offshore speed is often a useful first estimate for long-distance arrival forecasting, though real warning models use detailed bathymetry and numerical simulation rather than a single constant depth.
This calculator is intentionally a teaser model. It assumes the long-wave approximation, ignores complicated coastal geometry, neglects dissipation and reflection, and does not model nonlinear breaking. In university-level or research treatments, tsunami behavior is described with more complete shallow-water equations or Boussinesq-type models that include momentum balance, varying bathymetry, and sometimes dispersion. Near breaking, fully nonlinear effects become important as well.
Even so, the simple formulas here teach the central ideas clearly: deeper water gives faster motion, shallower water slows the wave, and shoaling can increase amplitude significantly as the tsunami approaches the coast. Those three ideas are at the heart of why distant offshore disturbances can turn into serious coastal hazards.