Stating point source in wave equation

[dzabakh]

Hello everybody!
I am trying to solve the wave equation in 3D
[latex]\frac{\partial^2 p}{\partial t^2} - c^2 \Delta p = \delta(\vec{r}-\vec{r_0}) \delta(t-t_0),[/latex]
where [latex]\vec{r_0}[/latex] is the radius-vector of the point source and [latex]t_0[/latex] is the moment when the source radiates a pulse, with Dirichlet boundary conditions.

I am solving the equation with finite-difference method in time domain. The steps by time are done by Runge-Kutta 4. The source is put into one node and the time profile of the source is step-like. I made it step-like (see Pic. 1)



because a point source should radiate a time-derivative of the incoming signal
[latex]p(\vec{r},t) = \frac{\rho_0 S}{4\pi r} \frac{dv}{dt}(t-r/c_0)[/latex]
and I want to get a delta-like signal from my source.

When I solve the equation, what I get is an impulse on the receiver, see Pic. 2.



This impulse is o'k but the amplitudes of such impulses fade incorrectly with distance. As I understand, it should be [latex]\sim 1/r[/latex] but in fact the amplitudes fade [latex]\sim e^{-\alpha r}[/latex], see Pic. 3.



I think I have stated the source incorrectly. Maybe someone knows how to do it right? What can also be a reason? Is this some kind of grid dispersion?

Thank you in advance.

[Klaynos]

What are your boundary conditions? Can you clarify that on the x axis it is distance from source at a given time?

[dzabakh]

What are your boundary conditions? Can you clarify that on the x axis it is distance from source at a given time?

I have Dirichlet boundary conditions. Yes, the x axis is c*t, where c is the speed of sound and t is the time.

[Klaynos]

And the y axis is something like E^2?

 

I'm not sure that you want dirchlet boundaries for your box. As that might give some unusual effects. I was expecting PMLs or similar.

 

It has been 5 years...

[dzabakh]

And the y axis is something like E^2?

 

I'm not sure that you want dirchlet boundaries for your box. As that might give some unusual effects. I was expecting PMLs or similar.

 

It has been 5 years...

Well, I cut the unusual effects off by a time window. I just want to send a pulse and the receive a diffracted pulse, I'm not much interested in the boundary conditions.

[Klaynos]

But if your boundary conditions are forcing a field value the. They will alter what you measure inside the box.

 

What package are you using? It was a few years ago but i found meep very accessible.

[dzabakh]

But if your boundary conditions are forcing a field value the. They will alter what you measure inside the box.

 

What package are you using? It was a few years ago but i found meep very accessible.

My code is written in Matlab. No, I think the boundary conditions shouldn't affect the impulse coming straightly from the source to the receiver. This pulse doesn't know nothing about the boundaries. I cut off the other pulses reflected from the boundaries with time window.