# 3D Partial Differential Equation

• Mar 14th 2009, 02:25 PM
BenWong
3D Partial Differential Equation
A rectangular plate of sides a and b has its edges fixed in the xy plane and set into transverse vibration. If the initial displacement is f(x,y) and the initial velocity is g(x,y), find the displacement u(x,y,t).
Does anybody know how to solve this partial differential equation?
• Mar 14th 2009, 02:54 PM
Jester
Quote:

Originally Posted by BenWong
A rectangular plate of sides a and b has its edges fixed in the xy plane and set into transverse vibration. If the initial displacement is f(x,y) and the initial velocity is g(x,y), find the displacement u(x,y,t).
Does anybody know how to solve this partial differential equation?

It helps if you give us the PDE. Is it

$\displaystyle u_{tt} = c^2 \left( u_{xx} + u_{yy}\right)$ or $\displaystyle u_{tt} + c^2 \left( u_{xxxx} + 2 u_{xxyy} + u_{yyyy}\right)=0\;?$
• Mar 14th 2009, 03:16 PM
BenWong
I believe it's the first one you had, although I'm not sure. The textbook I'm reading just has the physical setting.
Let's assume the first.
• Mar 14th 2009, 04:01 PM
Jester
Assume the usual separation of variables $\displaystyle u = T(t)X(x)Y(y)$. Substitute and separate into 3 ODE, i.e.
$\displaystyle \frac{1}{c^2} \frac{T''}{T} = \frac{X''}{X} + \frac{Y''}{Y}$

so

$\displaystyle \frac{X''}{X} = \lambda_1$, $\displaystyle \frac{Y''}{Y} = \lambda_2$, and $\displaystyle \frac{1}{c^2} \frac{T''}{T} =\lambda_1 + \lambda_2$

Now, bring in the BC's for the first two ODE's giving (details omitted)

$\displaystyle X = c_1 \sin \frac{ \pi\, x }{a},\;\;\;Y = c_2 \sin \frac{ \pi\, y}{b}$

$\displaystyle \frac{1}{c^2} \frac{T''}{T} = - \left( \frac{\pi}{a} \right)^2 - \left( \frac{\pi}{b} \right)^2$