Algebra Applications: Tsunamis

Narrator: On July 17, 1998 a tsunami struck the northern

coast of the island of New Guinea.

The resulting waves quickly overwhelmed the coast.

Villages were destroyed, thousands died.

A tsunami is the result of an undersea earthquake.

The massive energy from the earthquake is transferred to

the water.

Whenever a body of water is disturbed a wave is formed, as

you can see from a pebble dropped in a pond.

In fact, the earthquake itself is made up of seismic waves,

which are vibrations in the Earth's crust.

So the energy from the seismic wave is transferred to the

energy in the tsunami wave.

The New Guinea earthquake hit very close to the coastline.

This meant there was little time to warn the population

about the tsunami.

This also meant that the size of the tsunami was large.

Throughout the Pacific there are detection devices that

monitor the oceans to check for earthquakes in the ocean

and possible tsunamis.

These devices consist of a buoy on the surface of the

water and an earthquake monitor on the ocean floor.

Together these devices provide an early warning system

for tsunamis.

Earthquake magnitude is a logarithmic measure, but the

amount of energy that an earthquake releases

is exponential.

This table shows the dramatic increases in energy output for

each half value of magnitude.

The energy values are equivalent to tons of

TNT exploding.

As you can see for low values of magnitude the amount of

energy is relatively small, but watch what happens when

the magnitude is greater than four.

The magnitude of the New Guinea earthquake was seven

point zero.

The amount of energy for this magnitude of earthquake is

equivalent to a thermonuclear explosion.

In other words, the earthquake that caused the New Guinea

tsunami was the same as an underwater nuclear explosion

causing a tsunami.

Let's look at a graph of the data to see how dramatic the

change in energy is.

Turn on the TI-Nspire.

Create a new document.

You may need to save a previous document.

Create a spreadsheet window.

Move to the top of column A.

Input the label x.

Now move to the top of column B and input the label y.

Move to cell A1 and input the data from the table.

Pause the video to input the data.

After you have input the data create a scatter plot.

Press the HOME key and select the data and

statistics option.

You'll see a scattering of points.

Use the nav pad to move to the x-axis.

Press ENTER to see a list of options.

Use the nav pad to select X and press ENTER.

Now move the nav pad to the y-axis.

Press ENTER to see the list of options.

Select Y and press ENTER.

Your screen should look like this.

Notice that this graph, while it shows all the data points,

requires that the y-axis go from zero to

thirty-two billion.

With this large a window most of the coordinates form a

flat line, and two points show the explosive growth that is

typical of exponential graphs.

It would be better to have a more reasonably sized display

window and have the points not collapsed near the x-axis.

This is why graphs of this type use a logarithmic scale.

Press CONTROL and the LEFT ARROW to return to

the spreadsheet.

Move the cursor to the top of column C and add the column

heading log scale.

Press ENTER.

The formula cell is now highlighted.

Input this formula into the cell.

What this formula does is calculate the logarithm of

every value in column B.

Press ENTER.

Now go back to the stat window you created.

As before, use the nav pad to move the pointer to

the y-axis.

Press ENTER to bring up the list of options.

This time select log scale.

Your screen should look like this.

Notice that the range of values on the y-axis is much

better, allowing you to use a better distribution of points.

One thing to notice about a logarithmic scale is that the

graph of exponential data is linear.

There is a simple explanation for this.

Logarithms and exponents are inverse functions.

This means that the log of an exponent leads to a

linear term.

Finally, the relationship between energy and magnitude

is shown in this equation.

Notice that energy, E, is expressed in a

logarithmic scale.

If you replace the term log e with f of X then this equation

becomes a linear equation.

The slope, one point five, is the slope of the scatter plot

you created earlier.

This is an example of linearizing data that would

otherwise be cumbersome to graph and analyze.

This technique is important when dealing with

exponential data.

And this is more than just a cosmetic change for equations.

In a life or death situation with tsunamis and earthquakes,

making quick calculations and creating readable graphs makes

for timely decision making.

[Music]