- 4.2.1 The physical problem
- 4.2.2 The mathematical problem
- 4.2.3 Dealing with the boundary conditions
- 4.2.4 The final solution

4.2 Extension to finite regions

If is restricted by finite boundaries, the D'Alembert solution
does not really apply. To use it anyway, we must somehow extend the
problem to a doubly infinite -range without boundaries. But our
solution without boundaries should still *satisfy* the boundary
conditions for the finite range. That is often possible by clever use
of symmetry. An example can clarify that.

4.2.1 The physical problem

The problem is to find the pressure for sound wave propagation in a
tube with one end closed and one end open:

4.2.2 The mathematical problem

- There is a finite domain given by
- There is an unknown pressure to be found from
the wave equation

- The constant is the speed of sound . The equation is only valid for normal acoustics in which the gas velocities are much less than the speed of sound.
- This equation is second order in time so it needs two initial
conditions. They are:

- This equation is second order in space so it needs two boundary conditions. Given are one homogeneous Neumann boundary condition at and one homogeneous Dirichlet condition at .

4.2.3 Dealing with the boundary conditions

The D'Alembert solution applies to an infinite domain
. So to use the D'Alembert solution, the given
initial conditions, that are valid for must be extended to
all . In other words, functions and must be
converted into functions and that have
values for all . Of course, in the interval , they must
stay the same as and . Assume now for example that
looks as sketched below:

You might think that you could now simply take to be zero for all outside the range of the pipe. The corresponding D'Alembert solution will satisfy the wave equation everywhere, including inside the pipe . That is good, because the wave equation must indeed be satisfied. Unfortunately, the solution you get that way will not satisfy the boundary conditions at and . So it will still be wrong.

*You must select the extension of to all
so that the correct boundary conditions become automatic.*

The way to do it is as follows:

- To make the boundary condition at automatic,
create
*symmetry*around . Symmetric functions have zero derivative at the symmetry point. - To make the boundary condition at automatic,
create
*antisymmetry*(odd symmetry) around . Anti-symmetric functions have zero derivative at the symmetry point.

The process is shown for below:

Create the extended function or the same way.

It is OK if you get kinks or discontinuities in your functions and while creating (anti)symmetry. This happens when and/or does not satisfy the given boundary conditions. While then or may not have a unique value at the initial time, that problem will disappear when the time becomes greater than zero.

4.2.4 The final solution

This is pretty easy to evaluate for simple functions and . You will have fun doing it.

In the range , the found solution is exactly the same as for the finite pipe! The solution outside that range can simply be ignored.