Physical interpretation of Neumann-Dirichlet conditions

[A.Magnus]

I am working on a PDE problem like this:

Consider the wave equation with homogeneous Neumann-Dirichlet boundary conditions:

$\begin{align} u_{tt} &= c^2U_{xx}, &&0<x<\mathscr l, t > 0\\ u_x(0, t) &=u(\mathscr l, t) = 0, &&t > 0\\ u(x, 0) &=f(x), &&0<x< \mathscr l\\ u_t(x, 0) &=g(x), &&0<x< \mathscr l \end{align}$
​

(a) Give a physical interpretation for each line in the problem above.
(b) State the eigenvalue problem for ...
(c) ...
(d) ...

I am posting this asking for help on answering (a) since I do not have background whatsoever in either engineering or physics. I know how to work out the rest of questions after (a), since they are all math questions.

Thank you very much for your time and help.

 

[Dick]

I am working on a PDE problem like this:

Consider the wave equation with homogeneous Neumann-Dirichlet boundary conditions:

$\begin{align} u_{tt} &= c^2U_{xx}, &&0<x<\mathscr l, t > 0\\ u_x(0, t) &=u(\mathscr l, t) = 0, &&t > 0\\ u(x, 0) &=f(x), &&0<x< \mathscr l\\ u_t(x, 0) &=g(x), &&0<x< \mathscr l \end{align}$
​

(a) Give a physical interpretation for each line in the problem above.
(b) State the eigenvalue problem for ...
(c) ...
(d) ...

I am posting this asking for help on answering (a) since I do not have background whatsoever in either engineering or physics. I know how to work out the rest of questions after (a), since they are all math questions.

Thank you very much for your time and help.

Well, it's a wave equation in one dimension. c is the propagation speed of the wave. The first equation is just the wave propagation equation away from the boundary. The others are just boundary conditions. Try to express them in words, if u represents the amplitude of the wave

 

[A.Magnus]

Well, it's a wave equation in one dimension. c is the propagation speed of the wave. The first equation is just the wave propagation equation away from the boundary. The others are just boundary conditions. Try to express them in words, if u represents the amplitude of the wave

Thank you.