General review & continuation of Chapter 2 computer lab

effortstom.h.wilson

tom.wilson@mail.wvu.edu

Department of Geology and GeographyWest Virginia University

Morgantown, WV

Problems 1-20

Tom Wilson, Department of Geology and Geography

Don’t forget to hand in problems 1-20 before leaving today

Objectives for the day

Tom Wilson, Department of Geology and Geography

• Look over the list of mathematical models used to represent geological relationships

•Introduce another class of mathematical functions used to represent geological and other observations in mathematical form.

• Continue using computer to visualize mathematical models, develop comprehensive solutions and help gain a broader understanding of our data and their implications.

Mathematical models we’ve reviewed

Tom Wilson, Department of Geology and Geography

• Linear

• Quadratic

• General polynomial (order n)

• Power laws

• Exponential (or allometric)

• Logarithmic

These mathematical functions allow us to represent a variety of geological data in mathematical form.

These models allow us to quantify our observations and make predictions about future behaviors.

Using sines and cosines to represent arbitrary functions. The Fourier series: a weighted sum of sines and cosines

• Periodic functions and signals may be expanded into a series of sine and cosine functions

0 1 1

2 2

3 3

( ) cos sin

cos 2 sin 2

cos3 sin 3

... +...

f t a a t b t

a t b t

a t b t

Note that the Fourier series shares some similarity to the order-n polynomial

Tom Wilson, Department of Geology and Geography

0 1 1

2 2

3 3

( ) cos sin

cos 2 sin 2

cos3 sin 3

... +...

cos sinn n

f t a a t b t

a t b t

a t b t

a n t b n t

20 1 2( ) ... n

nf t a a t a t a t

Simulate the step using a sum of sines and cosines. An impossible task?

Tom Wilson, Department of Geology and Geography

Fourier series• Try the excel file step2.xls (see link on class page)

01

( ) cos sinn nn

f t a a n t b n t

The step function is defined by the Fourier series shown below

Tom Wilson, Department of Geology and Geography

1 1 1 ( ) sin( ) sin(3 ) sin(5 ) sin(7 ) ....3 5 7the step function f t t t t t

Function of t , x, or other independent

variable.

This applet is fun to play with & educational too.

Experiment with http://www.falstad.com/fourier/

Fourier series

• The Fourier series can be expressed more compactly using summation notation

01

( ) cos sinn nn

f t a a n t b n t

You’ve seen from the forgoing example that right angle turns, drops, increases in the value of a function

can be simulated using the curvaceous sinusoids.

Another function to add to your tool kit.

Problem 2.11 – questions?Due Tuesday

Tom Wilson, Department of Geology and Geography

Problem 2.12 – Questions/Review?

Tom Wilson, Department of Geology and Geography

A quick look ..

Calculate the concentration of an element at 50% crystallization: i.e. F=0.5 (the liquid fraction).

In the excel solution we compute C for values of F ranging from 0 to 1 in intervals of 0.05.

Text problems 2.11 and 2.12

Tom Wilson, Department of Geology and Geography

The computer analysis and questions associated with problems 2.11 and 2.12 include answers to the text questions associated with these two problems (page 40 of text). Nothing additional should be required.

Before continuing, let’s take a look at a couple additional problems (turn in before leaving)

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Finish up the day with a look at problem 2-13 using Excel

Tom Wilson, Department of Geology and Geography

A couple questions for general discussion

Tom Wilson, Department of Geology and Geography

2. Solve the above for a.

Let’s get started on problem 2.13

Tom Wilson, Department of Geology and Geography

Value of a at time =0 using the formula =a0*EXP(-lam*A2)

Assign variable names lam and a0 for these two constants.

Calculate ln(a) (column B, cell B2) as =LN(a0)-lam*A2

Problem 2.13

Tom Wilson, Department of Geology and Geography

See page 39 of the lab guide for detailed check list.

•Statement of problem•Submit two excel plots•Label plots showing values in question•Present calculations of t(a=100) as a cross check on your computer work•State result

•Finish up computer problems 2.11 and 2.12. Due next Tuesday

•Continue working through the computer lab problem 2.13

• Bring questions to class on Tuesday

• Hand in the group problems before leaving (i.e determine the intercept and Co)

• Again - Excel problems 2.11 and 2.12 are due next time - See page 26 of lab guide (and reminder handout) for presentation format.

To do list …