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.
