Timeline for Finite speed of propagation of wave equation
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 13, 2018 at 22:03 | comment | added | der noyoon | Can anyone of you explain me from where the term(at the boundary:S_r-t ) with -sign appeared after one differentiate E ? Thanks | |
| Jun 27, 2014 at 1:57 | comment | added | Christian Remling | Oh, so it seems I somewhat misunderstood the question (sorry). But can't we just argue as follows: Suppose that $u(t=0)$, $u_t(t=0)$ have compact support inside $\Omega$. Solve the wave equation in $\mathbb R^n$ with these initial conditions (and extended to the whole space in the obvious way, by setting them equal to zero outside $\Omega$). Then $u=0$ on $\partial\Omega$ for a while by FPS on the whole space. But this function, restricted to $\Omega$, solves your problem as well (as long as $u=0$ on $\partial\Omega$). | |
| Jun 27, 2014 at 1:17 | comment | added | student | I am more interested in the physical significance of the term FPS. In other words, I want to make sure that if the support of the initial condition is away from the boundary, for some small time $t$ the solution will still be zero on the boundary. Can you please shed some light on this? This was the basic reason behind my question.... | |
| Jun 26, 2014 at 21:49 | vote | accept | student | ||
| Jun 26, 2014 at 21:48 | vote | accept | student | ||
| Jun 26, 2014 at 21:49 | |||||
| Jun 26, 2014 at 21:48 | comment | added | student | But can you claim finite propagation speed if your argument does not work on balls centered at the boundary? | |
| Jun 26, 2014 at 21:28 | comment | added | Christian Remling | (1) I think an easy argument is to write the integral in spherical coordinates, then the radius of $B$ only affects the integration in the radial variable and the claim becomes the fundamental theorem. (2) I don't think this can work, except perhaps in special situations (such as boundary is a half plane locally). | |
| Jun 26, 2014 at 20:31 | comment | added | student | Sorry for the late reply. This seems convincing enough. I have two questions: 1. Can you give a reference to the differentiation under the integration formula where the boundary is dependent on the variable? I know the formula you state is right, I am just looking for a reference. 2. I think the calculation you give should work even when the center of the ball $B_{r−t}$ is on the boundary. Is that correct? | |
| Jun 19, 2014 at 19:21 | history | edited | Christian Remling | CC BY-SA 3.0 |
deleted 21 characters in body
|
| Jun 19, 2014 at 17:01 | history | answered | Christian Remling | CC BY-SA 3.0 |