FDEs
Moderator: Mike Everman

 Posts: 4930
 Joined: Fri Oct 31, 2003 7:25 am
 Antipspambot question: 0
 Location: santa barbara, CA
 Contact:
I remain intimidated. ha
Mike
__________________________
Follow my technical science blog at: http://mikeeverman.com/
Get alerts for the above on twitter at: http://twitter.com/mikeeverman
__________________________
Follow my technical science blog at: http://mikeeverman.com/
Get alerts for the above on twitter at: http://twitter.com/mikeeverman

 Posts: 3716
 Joined: Tue Dec 07, 2004 6:51 pm
 Antipspambot question: 0
 Location: 41d 1' N 80d 22' W
the Makings of a Right Moving Pulse (re: FDEs)
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 .
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 .

 Posts: 3716
 Joined: Tue Dec 07, 2004 6:51 pm
 Antipspambot question: 0
 Location: 41d 1' N 80d 22' W
... linear wave equation (re: FDEs)
Now, some of you are probably wondering why study the
linear wave equation.
Well, ...
linear wave equation.
Well, ...

 Posts: 3716
 Joined: Tue Dec 07, 2004 6:51 pm
 Antipspambot question: 0
 Location: 41d 1' N 80d 22' W
... advective transport (re: FDEs)
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.
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.

 Posts: 3716
 Joined: Tue Dec 07, 2004 6:51 pm
 Antipspambot question: 0
 Location: 41d 1' N 80d 22' W
... oneway wave equation (re: FDEs)
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 firstorder hyperbolic equation. In this
context the equation is sometimes called the oneway wave equation
because it models waves propagating in one particular direction.
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 firstorder hyperbolic equation. In this
context the equation is sometimes called the oneway wave equation
because it models waves propagating in one particular direction.

 Posts: 3716
 Joined: Tue Dec 07, 2004 6:51 pm
 Antipspambot question: 0
 Location: 41d 1' N 80d 22' W
... DiffEQ (re: FDEs)
Let's try to solve an ordinary DiffEQ, first.

 Posts: 3716
 Joined: Tue Dec 07, 2004 6:51 pm
 Antipspambot question: 0
 Location: 41d 1' N 80d 22' W
... spreadsheet calc (re: FDEs)
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.
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.

 Posts: 3716
 Joined: Tue Dec 07, 2004 6:51 pm
 Antipspambot question: 0
 Location: 41d 1' N 80d 22' W
... DiffEQ (re: FDEs)
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 crosssectional dimensions.
The deflection, y, is measured from the central axis; +tive down.
The station, x, is measured from the left support.
~~~~~~~~~~~~
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 crosssectional dimensions.
The deflection, y, is measured from the central axis; +tive down.
The station, x, is measured from the left support.