FDEs

[WebPilot]

... is an acronymn for:

1. Finite Difference Equations or

2. my initials, whichever.

There was a time when the following intimidated me ... those days are
over.

(c.t.)

[WebPilot]

This is a test.

Notice the courant number is only 0.4 .

(c.t.)

[Mike Everman]

I remain intimidated. ha

[WebPilot]

Hmmm, I can identify.

Here, a Courant number of 0.8 was used to generate the plots.

Comparing this with the last plot, notice an overall shift to the
right and the 'smearing' of the numerical solution is diminishing.

(c.t.)

[WebPilot]

Notice here that with a value of Cn=1.0, the numerical solution
converges with the exact, analytical solution. This is the value
I shall use in all subsequent calculations for this problem.

(c.t.)

[WebPilot]

Heretofore, I have been showing images after 20 time steps. Now,
this is a plot of the first 2 timesteps, using CN=1.0.

The pulse is starting to move to the right ...

(c.t.)

NOTE: 'x' is a dimensionless parameter which signifies the
station along the tube. For example, a point 10" from the left end of
a 100" tube would have an 'x' value of 10/100 = 0.10 .

[WebPilot]

Here, all 20 time steps are plotted. The pulse is definitely moving to
the right.

(c.t.)

[WebPilot]

Now, some of you are probably wondering why study the
linear wave equation.

Well, ...

[WebPilot]

Advective transport refers to a substance being carried along
with fluid motion. For example, consider a contaminant being
advected downstream with some fluid flowing through a one-
dimensional pipe at a constant velocity, u.

The concentration profile (or waveform) propagates with con-
stant speed, u and unchanged shape. In this context, the
linear wave equation is generally called the advection equation.

[WebPilot]

The phenomenon of wave motion is observed in its most basic form
if we model a sound wave traveling down a tube of gas or through an
elastic solid. In this case the molecules of the gas or solid barely move,
and yet a distinct wave can propagate through the material with its
shape essentially unchanged over long distances, and at a speed c
(the speed of sound in the material) that is much larger than the velocity
of material particles.

A sound wave propagating in one direction (to the right with speed c > 0)
can be modeled by the equation

wt(x, t) + c wx(x, t) = 0,

where w(x, t) is an appropriate combination of the pressure and particle
velocity.

This again has the form of a scalar first-order hyperbolic equation. In this
context the equation is some-times called the one-way wave equation
because it models waves propagating in one particular direction.

[WebPilot]

Let's try to solve an ordinary DiffEQ, first.

[WebPilot]

It is my intention to use a spreadsheet for this calculation.

Solving the ordinary diffEQ first, will prepare the interested
reader to solve the linear wave equation, next.

I am going to use Excel, but any other spreadsheet may
do. So, follow along and see if you can get it to work in
the one you use, too.

[GRIM]

Hey Web Pilot
I see you dont get many replies to your posts , rest assured I for one am eagerly awaiting the next instalment , fascinating stuff , and the car and bike photos are excellent.

keep em coming and thanks.

[WebPilot]

You are quite welcome, GRIM.

~~~~~~~~~~~~

As a start, I've decided to use the differential equation describing
the deflection of a simply supported beam, due its own weight.

The derivation is in any engineering mechanics book.

(c.t.)

NOTE:

The beam's length is >> greater than its cross-sectional dimensions.

The deflection, y, is measured from the central axis; +tive down.

The station, x, is measured from the left support.

[WebPilot]

Uniformly Loaded Beam ...

(c.t.)