Sturm-Liouville-Wave equation

[renegade05]

Trying to do this one...

A model for the transverse vibrations of a stretched string with variable density ρ and tension τ (both continuous and strictly positive on the closed interval [0,l]):


PDE: \(\displaystyle ρ(x)u_{tt} − [τ (x)u_x]_x = 0, 0 < x < l, t > 0,\)


BC: \(\displaystyle u(0, t) = 0, u_x(l, t) + A_l u(l, t) = 0 \quad(for \quad all \quad t > 0)\)


One end of the string is fixed, and at the other end the string exchanges some energy with the endpoint). If u(x, t) = X(x)T(t) is to be a separated solution of the PDE, find the ODEs that T(t) and X(x) must satisfy. If the BC are also to be satisfied, verify that X(x) must then be a solution of a Sturm-Liouville problem. In particular, verify explicitly that the boundary conditions for the boundary-value problem for X(x) are symmetric.

So

if \(\displaystyle u(x, t) = X(x)T(t) \)

\(\displaystyle u_{tt} = X(x)T''(t) \)

\(\displaystyle u_{x} = X'(x)T(t)\)

\(\displaystyle u_{xx} = X''(x)T(t)\)

So:

PDE: \(\displaystyle ρ(x)u_{tt} − [τ (x)u_x]_x = ρ(x)u_{tt} - τ_x(x)u_x- τ(x)u_{xx} \)

\(\displaystyle = ρ(x)X(x)T''(t) - τ_x(x)X'(x)T(t) - τ(x)X''(x)T(t)\)

\(\displaystyle \rightarrow ρ(x)X(x)T''(t) = (τ_x(x)X'(x) + τ(x)X''(x))T(t)\)

\(\displaystyle \rightarrow \frac{T''(t)}{T(t)} = \frac{τ_x(x)X'(x) + τ(x)X''(x)}{ρ(x)X(x)} = -\lambda\)

\(\displaystyle (1)T''(t)+\lambda T(t) = 0 \)

\(\displaystyle (2) τ(x)X''(x) + τ_x(x)X'(x) +\lambda ρ(x)X(x)=0\)

I think up to this point is correct? Not 100% confident though.

So I need to answer these questions:

(a) Find the ODE's that T(t) and X(x) must satisfy - I think the ODE's are (1) (2) above?

(b) If the BC are also satisfied, verify X(x) must be a solution of a S-L problem.

(c)In particular, verify that the boundary conditions for the boundary - value problem for X(x) are symmetric.

Think I got (a) but stuck on (b) and (c)

 

[Ishuda]

...
So I need to answer these questions:

(a) Find the ODE's that T(t) and X(x) must satisfy - I think the ODE's are (1) (2) above?

(b) If the BC are also satisfied, verify X(x) must be a solution of a S-L problem.

(c)In particular, verify that the boundary conditions for the boundary - value problem for X(x) are symmetric.

Think I got (a) but stuck on (b) and (c)

Yes, you have (a) and, since you are given τ(x) and ρ(x) as positive, you can write the general form of the S-L problem as your equation (2):
\(\displaystyle (2) τ(x)X''(x) + τ_x(x)X'(x) +\lambda ρ(x)X(x) = 0\)
or
\(\displaystyle (τ(x)X'(x))' + \lambda ρ(x)X(x) = 0\)
generally with boundary conditions. Thus you have (b).

I'll need to wait a bit before I tackle (c) as I have to do something else right now.

 

[Ishuda]

...
(c)In particular, verify that the boundary conditions for the boundary - value problem for X(x) are symmetric.

Looking at this further, I'm not sure what you mean by "the boundary conditions for the boundary - value problem for X(x) are symmetric." unless you mean both u(\(\displaystyle l,\, t_0\, -\, t\)) and u(\(\displaystyle l,\, t_0\, +\, t\)) satisfy the boundary conditions which they obviously do for all \(\displaystyle t_0\)since the boundary conditions are independent of T.

 

[renegade05]

Looking at this further, I'm not sure what you mean by "the boundary conditions for the boundary - value problem for X(x) are symmetric." unless you mean both u(\(\displaystyle l,\, t_0\, -\, t\)) and u(\(\displaystyle l,\, t_0\, +\, t\)) satisfy the boundary conditions which they obviously do for all \(\displaystyle t_0\)since the boundary conditions are independent of T.

Well the BC are:

\(\displaystyle u(0, t) = 0, u_x(l, t) + A_l u(l, t) = 0 \quad(for \quad all \quad t > 0)\)

So I am assuming I have to use these somewhere in the problem.