Thread: Wave Equation using seperation of variables

Hello i was wondering if someone could help me out and show me how to solve the following heat wave problem.We dont have a book for this class and i couldnt make it to my last class so i dont knoow how to go about this question

Solve the wave equation using seperation of variables and show thte the solution reduces to D'Alembert's solution

$\displaystyle \mu_{tt} = c^2\mu_{xx} $
$\displaystyle \mu(0,t) = 0$
$\displaystyle \mu(L,t) = 0$
$\displaystyle \mu(x,0) = f(x)$
$\displaystyle \mu_t(x,0) = 0$

Originally Posted by flaming

Hello i was wondering if someone could help me out and show me how to solve the following heat wave problem.We dont have a book for this class and i couldnt make it to my last class so i dont knoow how to go about this question

Solve the wave equation using seperation of variables and show thte the solution reduces to D'Alembert's solution

$\displaystyle \mu_{tt} = c^2\mu_{xx} $
$\displaystyle \mu(0,t) = 0$
$\displaystyle \mu(L,t) = 0$
$\displaystyle \mu(x,0) = f(x)$
$\displaystyle \mu_t(x,0) = 0$

I would feel better if you would at least show that you know what "separation of variables" is!

Assume $\displaystyle \mu(x,t)= X(x)T(t)$. Then $\displaystyle \mu_{tt}= XT^{''} $ and $\displaystyle \mu_{xx}= X^{''} T$ so the equation becomes $\displaystyle XT^{''} = c^2X^{''} T$. Dividing on both sides of the equation by XT gives $\displaystyle \frac{T^{''} }{T}= c^2\frac{X^{''} }{X}$. Since the left side depends only on t and the right side only on x, to be equal they must each equal a constant.

That gives two ordinary equations:
$\displaystyle c^2\frac{X^{''} }{X}= \lambda$
or
$\displaystyle c^2X^{''} = \lambda X$
and
$\displaystyle \frac{T^{''} }{T}= \lambda$
or
$\displaystyle T^{''} = \lambda T$.
You should be able to find the general solution to each of those, depending on $\displaystyle \lambda$ of course.
What must $\displaystyle \lambda$ be in order that X(0)= 0 and X(L)= 0?

I have a feeling I have repeated what your text book says!