# wave partial differential eqn

• Feb 11th 2009, 12:55 PM
PvtBillPilgrim
wave partial differential eqn
Can someone help me solve this: the vibrating string partial differential equation if an external force per unit mass proportional to displacement acts on the string.

So I have uxx - ku = (1/a^2)utt

How do I solve this? I've only seen it without external forces (here Pauls Online Notes : Differential Equations - Vibrating String). I've just started studying PDE's.

• Feb 13th 2009, 06:44 AM
Jester
Quote:

Originally Posted by PvtBillPilgrim
Can someone help me solve this: the vibrating string partial differential equation if an external force per unit mass proportional to displacement acts on the string.

So I have uxx - ku = (1/a^2)utt

How do I solve this? I've only seen it without external forces (here Pauls Online Notes : Differential Equations - Vibrating String). I've just started studying PDE's.

Do you have boundary and initial conditions to go with this PDE?
• Feb 13th 2009, 04:04 PM
Rincewind
Quote:

Originally Posted by danny arrigo
Do you have boundary and initial conditions to go with this PDE?

That's what I was wondering.

I assume that the vibrating string has fixed endpoints.

$u(0,t) = u(L,t) = 0$

But you need some initial conditions as well to determine all arbitrary constants.
• Feb 14th 2009, 05:07 PM
Jester
Quote:

Originally Posted by Rincewind
That's what I was wondering.

I assume that the vibrating string has fixed endpoints.

$u(0,t) = u(L,t) = 0$

But you need some initial conditions as well to determine all arbitrary constants.

It could also have a free endpoint where $u_x(0,t) =0,\;\; \text{or}\;\;\;u_x(L,t) = 0$.
• Feb 14th 2009, 05:44 PM
Rincewind
Quote:

Originally Posted by danny arrigo
It could also have a free endpoint where $u_x(0,t) =0,\;\; \text{or}\;\;\;u_x(L,t) = 0$.

Yup. Anything is possible. I wonder if PvtBillPilgrim will clarify. (Thinking)
• Feb 16th 2009, 05:34 PM
PvtBillPilgrim
Quote:

Originally Posted by Rincewind
That's what I was wondering.

I assume that the vibrating string has fixed endpoints.

$u(0,t) = u(L,t) = 0$

But you need some initial conditions as well to determine all arbitrary constants.

I want these conditions.
Basically, u(x,0)=f(x)
du/dt(x,0)=g(x) where this is a partial derivative
u(0,t)=0=u(L,t)
How does the -ku play into the final solution? It's solved in the website I gave in the original post, but with no external forces.
• Feb 17th 2009, 03:23 PM
Jester
Quote:

Originally Posted by PvtBillPilgrim
I want these conditions.
Basically, u(x,0)=f(x)
du/dt(x,0)=g(x) where this is a partial derivative
u(0,t)=0=u(L,t)
How does the -ku play into the final solution? It's solved in the website I gave in the original post, but with no external forces.

Use the usual separation of variables $u = T(t) X(x)$ so your equation separates

$\frac{T''}{a^2 T} = \frac{X'' - k X}{X} = c$

From the second

$X'' - k X = c X$ or $X'' + \omega^2 X = 0$ so $c = - k - \omega^2$. I'm thinking you can take it from here.