# Wave equation with inhomogeneous boundary conditions

1. Mar 27, 2009

### josh146

I'm reposting this because there was a problem with the title/LaTeX last time.

1. The problem statement, all variables and given/known data

Solve the wave equation (1) on the region 0<x<2 subject to the boundary conditions (2) and the initial condition (3) by separation of variables.

2. Relevant equations

(1) $\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}$

(2) $\frac{\partial u}{\partial x}(0,t)=1$ ; $\frac{\partial u }{\partial x}(2,t)=1$

(3) $\frac{\partial u}{\partial t}(x,0)=0$

3. The attempt at a solution

I've defined $\theta(x,t)=u(x,t)-u_{st}(x) = u(x,t)-x-h(t)$ where u_st is the steady state solution (4). I've used this to create a new PDE with homogeneous boundary conditions.

The PDE is:

$\frac{\partial^2 \theta}{\partial t^2} + h''(t)=c^2 \frac{\partial^2 \theta}{\partial x^2}$.

By subbing in $\theta=f(t)g(x)$ I get:

$f''(t)g(x)+h''(t)=c^2 f(t) g''(x)$

I'm not sure how to transform this into two ODEs. Can someone help?

(4): The solution to $\frac{\partial^2 u}{\partial x^2} = 0$ subject to (2).

2. Mar 27, 2009

### quZz

I think you just need to introduce new function v: u(x,t) = v(x,t) + x. The equation for v(x,t) will be the same but boundary conditions will become homogeneous =) and btw, you are missing one more initial condition like u(x,t=0) = ...

3. Apr 23, 2009

### Spoony

Im also stuck on this one, for me its the initial conditions and boundary conditions being partial derivitives that throws me.

Last edited: Apr 23, 2009
4. Apr 25, 2009

### Spoony

The eqn. given for boundary conditions is a partial derivitive so do i intergrate it to find u(x,t)
in which case is it that; u(o,t) = x and u(2,t) = x. so that.
u(x,t) = v(x,t) + x so that in both cases.... umm then i get stuck.

5. Apr 25, 2009

### Spoony

No-one at all?

6. Apr 26, 2009

### Spoony

So far ive got that:
$$U_{st} (x) = x$$
(a) Find the steady state solution, ust(x), by solving $$\frac{\delta ^{2} U_{st} }{\delta x^{2}} = 0$$ and applying the boundary
conditions. (You will only be able to determine one of the two arbitrary constants).

confused here, surely $$U_{st} (x) = x + c$$ (as boundary conditions given are derivitives)

then initial condition:

$$\frac{\partial u}{\partial t}(x,0)=0$$

given means that c = 0? :S

7. Apr 26, 2009

### Spoony

Latex is fail :(

So far ive got that:
U_st (x) = x
(a) Find the steady state solution, u_st(x), by solving (partial second derivitive u by x = 0) and applying the boundary
conditions. (You will only be able to determine one of the two arbitrary constants).

confused here, surely U_st (x) = x + c (as boundary conditions given are derivitives)

then initial condition:

(as above... in first post)

given means that c = 0? :S

8. Apr 27, 2009

### Spoony

Can no one point me in the right direction?

9. May 10, 2009

### guhan

Instead of integrating the BC, you may differentiate the solution you get and then apply the BC.