# 1d wave eqution (help urgent)

• Oct 19th 2012, 04:58 PM
princessmath
1d wave eqution (help urgent)
Attachment 25289

Basically, I have to use u = Real (f(x)e^-iwt) and sub it into the wave equation.

Simplify it etc,

then use say u= e^at to solve the characteristics equation.

However, I get stuck.

Can someone show me how to do it.

Thank you
• Oct 20th 2012, 04:55 AM
TheEmptySet
Re: 1d wave eqution (help urgent)
Quote:

Originally Posted by princessmath
Attachment 25289

Basically, I have to use u = Real (f(x)e^-iwt) and sub it into the wave equation.

Simplify it etc,

then use say u= e^at to solve the characteristics equation.

However, I get stuck.

Can someone show me how to do it.

Thank you

If you assume

$\displaystyle u(x,t)=f(x)e^{-i \omega t} \implies u_{tt}=(i\omega)^2f(x)e^{-i\omega t}$

and

$\displaystyle u_{x}= f'(x)e^{-i \omega t} \text{ and } u_{xx}=f''(x)e^{-i \omega t}$

Now just expand out the right hand side of the eqation

$\displaystyle \frac{\partial }{\partial x} \left( A(x) \frac{\partial u}{\partial x}\right) = \frac{\partial A}{\partial x}\frac{\partial u}{\partial x}+A(x)\frac{\partial^2 u}{\partial x^2}=A'u'+Au''$

Now just put all of this into the equation

$\displaystyle (i \omega)^2f(x)e^{-i \omega t} = \frac{c^2}{A}\left(A'f'e^{- i \omega t}+f''e^{-i \omega t} \right)$

Can you finish from here?
• Oct 20th 2012, 01:38 PM
princessmath
Re: 1d wave eqution (help urgent)
Yeah, that what I did. Btw you forgot the A(x) next to f''e^-iwt

You can cancel e^-iwt etc.

But I'm stuck on how do I find f(x).
• Oct 20th 2012, 01:50 PM
princessmath
Re: 1d wave eqution (help urgent)
In short, I'm stuck on how do I solve the second order differential equation.
• Oct 20th 2012, 02:21 PM
TheEmptySet
Re: 1d wave eqution (help urgent)
Quote:

Originally Posted by princessmath
In short, I'm stuck on how do I solve the second order differential equation.

The equation can be written in the form

$\displaystyle -\frac{d}{dx}\left( A(x)\frac{df(x)}{dx}\right)=\frac{\omega^2}{c^2}f( x)$

The is a Sturm-Liouville ODE (it is in its self-adjoint form). Does the equation have boundary conditions?
• Oct 21st 2012, 03:02 AM
princessmath
Re: 1d wave eqution (help urgent)
No, the question doesn't have any boundary conditions.

Just said to find the ode f(x).
• Oct 21st 2012, 03:09 AM
princessmath
Re: 1d wave eqution (help urgent)
How would you go about solving that pde?

See what I get is

A(x)F''(x) + A'(x)F'(x) + w^2/c^2 F(x) = 0

and then I try letting m = e^bx

then solving for the charateristic, but then I get confused :(