How Fast a Tsunami Travels

In the open ocean a tsunami races along as fast as a jet airliner, yet slows to the speed of a car near shore. Its speed is the square root of gravity times the water depth, so water four times deeper only doubles the speed.

How it works

Radical functions and radical equations

In one sentence

A radical function involves a root of the variable, such as $v = \sqrt{gh}$ for $h \ge 0$ . A radical equation is solved by isolating the root and raising both sides to the matching power, $h = v^2/g$ ; since squaring can introduce extraneous solutions, every candidate must be checked in the original equation.

General formula

{\htmlData{sym=0}{v}} = \sqrt{{\htmlData{sym=1}{g}}\,{\htmlData{sym=2}{h}}}, \qquad {\htmlData{sym=2}{h}} = \frac{{\htmlData{sym=0}{v}}^2}{\htmlData{sym=1}{{g}}}

Doubling the speed takes four times the depth: square roots climb by smaller and smaller steps.

What the symbols mean

$v$
Wave speed
How fast the tsunami travels, in meters per second.
$g$
Gravity
9.8 m/s².
$h$
Water depth
The depth of the ocean under the wave, in meters.

Tsunami waves are so long that they feel the whole depth of the ocean. Their speed is $v = \sqrt{gh}$ , with $g = 9.8$ m/s² and $h$ the depth in meters.
A square root grows quickly at first and then levels off: going from 1,000 m to 4,000 m deep only doubles the speed.
To find the depth that gives a certain speed, solve the radical equation $v = \sqrt{gh}$ by squaring both sides: $h = v^2/g$ .
Squaring can introduce false solutions, so check the answer in the original equation: depths must be positive and the speed is the positive square root.

Learn the math

RadicalsAlgebra · Lesson 3, Preliminaries Equations with RadicalsAlgebra · Lesson 15, Solving Equations and Inequalities