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

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- Oct 19th 2012, 04:58 PMprincessmath1d 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 AMTheEmptySetRe: 1d wave eqution (help urgent)
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 PMprincessmathRe: 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 PMprincessmathRe: 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 PMTheEmptySetRe: 1d wave eqution (help urgent)
- Oct 21st 2012, 03:02 AMprincessmathRe: 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 AMprincessmathRe: 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 :(