Tuesday, March 15, 2011

Tsunami Mathematics

Tropical Tsunami
Image Source: http://millicentandcarlafran.files.wordpress.com

During 2011, most of us viewed news reports of powerful and devastating Tsunami waves that were produced by a 9.0 magnitude Earthquake
off the shore of Japan.

A Tsunami is actually a series of waves. In the open ocean the waves are not high at all, and can pass under ships with no noticeable effect.

But Tsnamis can travel at the speed of jet aircraft (700 km/hr), and can stretch in length for hundreds of kilometres across the ocean.

As the waves approach land, their energy intensifies, and the height of the wave can increase to as high as 60 meters.

Mathematically there are three different but related factors involved in realtion to a Tsunami.


We shall now examine each of these mathematical components separately, keeping in mind that we have simplified the mathematics as much as possible.

Detailed mathematical equations of water waves look something like this:

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Image Source: http://www.oceanographers.net

These equations require super computers to help work on them.

To see what the full set of real equations look like, as well as a computer model of a Tsunami, check out the following web page:


We have simplified the mathematics greatly in the material which follows.

Let's start with Tsunami speed.

Mathematical Speed of a Tsunami

In the open Ocean the speed of a Tsunami is approximately:

  • Speed = The Square root of (9.8 x Water Depth)

The 9.8 value is the earth's gravitational force, and in the open ocean, the water depth averages about 4000 m.

Sorry Picture Not Found
Image Source: http://2.bp.blogspot.com

Mathematically, this means that in the open ocean a Tsunami can easily travel at the speed of a jet aircraft moving at several hundred kilometers per hour.

(Source: http://terrytao.wordpress.com/2011/03/13/the-shallow-water-wave-equation-and-tsunami-propagation/)

However, when the Tsunami approaches close to a shoreline, and the depth decreases to a few hundred meters, the speed slows down, (as per the above maths equation), to a few hundred km / hr.

As the Tsunami waves slow down, they move closer together and rise in height. They can reach heights up to 60 metres when they arrive on shore, with speeds of 250 km/hr, producing a huge destructive force.

Mathematical Height of Tsunami Waves

As the Tsunami approaches shore, the water depth decreases, causing the Tsunami to slow down, at a rate proportional to the square root of the depth.

Unfortunately, "wave shoaling" then forces the Amplitude (Height) to increase at the opposite rate of:

  • Height is proportional to 1 divided by square root of water depth.

Eg. If the water depth is 400m and then decreases to 4m, then we have a 1/20 Amplitude rising to a 1/2 Amplitude, which means the wave will suddenly become 10 times higher. So a 1m high wave one km from the shoreline, suddenly becomes 10 metres high as it gets to the shoreline.

The following diagram shows this Mathematics of the Tsunami height increasing:

Sorry Picture Not Found
Image Source: http://www.cnsm.csulb.edu

The following picture shows a 10 meter high Tsunami hitting the coast of Japan.

Sorry Picture Not Found
Image Source: http://www2.registerguard.com

Mathematical Power of Tsunamis

The POWER of any wave is mathematically related to the square root of how LONG the wave is.

  • Energy Power is proportional to the square root of wave Length.

(Source: http://plus.maths.org/content/os/issue34/features/tsunami/index)

Normal waves at a beach on a rough day might look something like this:

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Image Source: http://cdnimages.magicseaweed.com

Long Waves at an Ocean beach produce much more powerful "surf" waves, because they have much more side to side length:

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Image Source: http://cdnimages.magicseaweed.com

In Victoria Australia, beaches with Long waves that are very powerful include: Kilcunda, Woolamai, Venus Bay, and to a much lesser extent Inverloch.

It is important to realise that if a wave doubles its left to right Length, then its power will increase by a factor of the square root of 2 or around 41%.

So the increase in Power is not double when we double the length, but is around 1 and a half times more powerful.

Tsuanamis are waves that are hundreds of kilometers long, and so the energy power they unleash is thousands of times greater than that of even the biggest crushing surf or storm waves.

And how did these Tsunami waves become so long?

Well we need to remember that the earthquake that produced them unleashed a mammoth amount of energy into the ocean. And this energy produces waves which are hundreds of kilometres long. Waves that are much longer than the winds from any big storm could ever produce.

Sorry Picture Not Found
Image Source: http://www.washingtonpost.com

The following video shows the long Tsumani waves hitting Japan in March 2011.

This next video that shows the power of these huge Tsunami waves as they hit the shore of Japan in 2011.

As can be seen, the Tsunami is more like a huge surge of water, rather than one big crashing wave. Also evident in these videos is that there are several Tsunami waves coming one after each other. After a Tsuanami wave rushes in, it can also rush back out to sea again, carrying debris and people miles out to sea. In addition, it is quite possible for more giant waves to keep arriving, even an hour after the first one has hit.

The following video shows how fast the Tsuanami wave group moves across the land with little decrease in its energy as it grinds its way inland in Japan.

In this next video the Tsunami overcomes the protective sea wall that had been built around a Japanese town:

So that's a bit of simplified Mathematics about the massively damaging power of Tsunamis. Hopefully it adds to our understanding of the world around us, even if it is only useful the next time we happen to visit a surf beach. Let's hope none of us ever do any first hand investigation of the mathematics of a Tsunami wave.

Big Passy Wasabi

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