Posted by: Rouhollah Ramezani

Â« on: October 01, 2012, 09:00:03 pm Â»

A) is correct

B) definitely contains an error which is easy to fix. Why I know about error? -- solution of RR is not $0$ as $x=t$

C) Contains a logical error in the domain $\{âˆ’2t<x<2t\}$ (`middle sector`) which should be found and fixed. Note that the solution of RR there is not in the form $\phi(x+2t)+\psi(x-2t)$

RR deserves a credit but there will be also a credit to one who fixes it

So, for a wave equation with a propagation speed $c$ and moving boundary (with a speed $v$) there are three cases (we exclude exact equalities $c=\pm v$ ) -- interpret them as a piston in the cylinder:

- $-c<v<c$ The piston moves with a subsonic speed: one condition as in the case of the staying wall
- $v>c$ The piston moves in with a supersonic speed: no conditions => shock waves etc
- $v<-c$ The piston moves out with a supersonic speed: two conditions.

3D analog: a plane moving in the air. If it is subsonic then everywhere on its surface one boundary condition should be given but for a supersonic flight no conditions on the front surface, one on the side surface and two on the rear (with $\vec{v}\cdot \vec{n} >c$, $-c< \vec{v}\cdot \vec{n} <c$ and $\vec{v}\cdot \vec{n} <-c$ respectively where $\vec{v}$ is the plane velocity and $\vec{n}$ is a unit outer normal at the given point to the plane surface. The real fun begins at transonic points where $\vec{v}\cdot \vec{n} =\pm c$).

PS MathJax is not a complete LaTeX and does not intend to be, so it commands like \bf do not work outside of math snippets (note \bf); MathJax has no idea about \newline as it is for text, not math. For formatting text use either html syntax (in plain html) or forum markup

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