1D wave equation open boundary

[emirs]

I am trying to write a solver for a 1D wave equation in MATLAB, and I have run into interesting problem that I just can't find a way out of.

I start with the wave equation, and then discretize it, to arrive at the following,

U{n+1}(j)=a*(U{n}(j+1)-2*U{n}(j)+U{n}(j-1))+2*U{n}(j)-U{n-1}(j) (for (j=1...end-1))

I'm trying to simulate an open end string (perturbate it in the middle for example, and I want the wave to disappear on the border)

Usually it is done (correct me if I'm wrong) with applying zero gradient boundary condition, which in my case is:
U{n}(end+1)=U{n}(end-1)
and leads to:
U{n+1}(end)=a*(-2*U{n}(end)+2*U{n}(end-1))+2*U{n}(end)-U{n-1}(end);

Unfortunately this condition creates a partial reflection from the end which is supposed to be free.

Please help

Regards

 

[emirs]

Anyone?

 

[Ben Niehoff]

Well, there ARE partial reflections from a free end, I think...the energy cannot transfer into the surrounding air with perfect efficiency.

But to eliminate as much reflection as possible, you should look up "absorbing boundary conditions". These are a bit tricky to implement and it may take a fair bit of reading to get them right.

 

[emirs]

Thank you, it is what I was looking for. You hae put me on the right track.

 

[blue_raver22]

I understand the challenge you are facing in writing a solver for a 1D wave equation with an open boundary. It is important to carefully consider the boundary conditions in order to accurately simulate the behavior of the wave.

One potential solution to your problem could be to use an absorbing boundary condition. This involves introducing a damping term at the boundary that dissipates the energy of the wave as it reaches the end. This can help reduce the reflection from the boundary and create a more realistic simulation of an open end.

Another approach could be to use a perfectly matched layer (PML) boundary condition. PML is a technique commonly used in computational electromagnetics to simulate open boundaries. It involves adding an artificial layer around the boundary that absorbs the wave without reflecting it back. This can be a more complex approach, but it may provide more accurate results.

I would also recommend checking your discretization scheme and ensuring that it is stable and accurate. Sometimes, small errors in the discretization can lead to unexpected behavior at the boundaries.

In conclusion, there are various techniques that can be used to simulate open boundaries in a 1D wave equation. It is important to carefully consider the specific conditions and choose the most appropriate approach for your simulation. I hope this helps and good luck with your solver.