Thin Membrane/Wave propagation
The FEMWave is a win32 screen saver which
is based on the two dimensional wave equation:
utt = c2(uxx + uyy)
+ f(x,y)
As is always the case this equation is basic mathematical model
for many physical processes. The simplest example of course is that of
a thin elastic membrane (in a drum, speakers, etc) undergoing small vertical
deflections. However the same equation is used to model torsional waves
in long elastic bars.
Typically, the forcing term f(x,y) is absent and under special conditions
exact solutions can be obtained. These usually involve simple geometry e.g.
the whole space R2, a half-plane or simple boundary conditions.
In all other cases one either has to know the appropriate Green's functions
or has to solve the problem numerically.
This is what the screen-saver does. The domain (in the current version)
is the unit square. The numerical method used is the semi-discrete Finite
Element Method (FEM). As for the setup dialog - it is quite obvious for
every person that has dealt with this matter before. In the near future
I intend to write a detailed description.... As for now here is a brief
otuline:
- Mesh parameter: Selects the number of nodes in one spatial
direction (1/h).
- Time step: Tells the solver what time step to use. In
order to get reasonable results, h and dt must be of the same order.
- Forcing function: You can input any valid arithemetic expression,
as well as sin(), cos() and exp(). The variables will be interpreted as
follows: x1 for the first spatial variable (x), x2 for the second spatial
variable (y) and t for the time varible. You can also select a randomly
placed point source that oscilates as sin(t), that is f(x,y)=Dirac(x0,y0)sin(t)
and x0,y0
are moved to a new location every 100 time steps (with a probability of
1/3). Currently the Dirac delta is not understood by the functional analyzer,
so do not attempt to use it.
- Boundary conditions: You can type in any valid arithmetic
expression. The same rules apply for the variable names. Currently there
is a choice of three boundary conditions which produce (in my opinion) nice
visual resutls.
There is also a variety of color options which can be adjusted according
to one's one taste. Enjoy.