# Second Order Partial Differential Equation (Wave Eqn)

• March 29th 2009, 09:24 AM
harkapobi
Second Order Partial Differential Equation (Wave Eqn)
Hi,
i have this question:

$u_{tt}$ = $c^2u_{xx}$

Find u(x,t) the deviateion from equilibrium for a stretched string fixed at its ends x=0 and x= pi. Initial conditions u(x,0)=alpha(sin(x) + 0.2sin(3x)), $u_{t}$(x,0)= 0.

I know to use seperation of variables u(x, t) = F(x)G(t) giving me two ode's

G"(t) - sG(t) = 0
F"(x) - (s/ $c^2$)F(x) = 0 where s is a constant.

This gives the solutions:
F(x) = A1sin(nx)
G(t) = A2cos(nct) + B2sin(nct)

Now i'm stuck, how do I use the boundary and initial conditions to solve this? I know it has something to do with the fourier series?

Katy
• March 29th 2009, 09:43 AM
Chris L T521
Quote:

Originally Posted by harkapobi
Hi,
i have this question:

$u_{tt}$ = $c^2u_{xx}$

Find u(x,t) the deviateion from equilibrium for a stretched string fixed at its ends x=0 and x= pi. Initial conditions u(x,0)=alpha(sin(x) + 0.2sin(3x)), $u_{t}$(x,0)= 0.

I know to use seperation of variables u(x, t) = F(x)G(t) giving me two ode's

G"(t) - sG(t) = 0
F"(x) - (s/ $c^2$)F(x) = 0 where s is a constant.

This gives the solutions:
F(x) = A1sin(nx)
G(t) = A2cos(nct) + B2sin(nct)

Now i'm stuck, how do I use the boundary and initial conditions to solve this? I know it has something to do with the fourier series?