Boundary Value Problems/Lesson 6

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1D Wave Equation

Derivation of the wave equation using string model.

General form for boundary conditions.

${\alpha }_{11}u(a,t)+{\alpha }_{12}{u}_x(a,t)={\gamma }_1$ ${\alpha }_{21}u(b,t)+{\alpha }_{22}{u}_x(b,t)={\gamma }_2$

Wave equation with Dirichlet Homogeneous Boundary conditions.

${\alpha }_{11}u(a,t)=0$
${\alpha }_{21}u(b,t)=0$
In the homogeneous problem $u_{xx}-\frac{1}{c^2}{u}_{tt}=0$ with $u(0,t)=0$ , $u(L,t)=0$

Finding a solution: u(x,t)

Let $u(x,t)=X(x)T(t)$
then substitute this into the PDE.
$X^{″}T=\frac{1}{c^2}X{T}^{″}$
$\frac{X^{″}}{X}=\frac{1}{c^2}\frac{T^{″}}{T}=\mu$
Where $\mu$ is a constant that can be positive, zero or negative. We need to check each case for a solution.

Wave Equation with nonhomogeneous Dirichlet Boundary Conditions

In the homogeneous problem $u_{xx}-\frac{1}{c^2}{u}_{tt}=0$
${\alpha }_{11}u(x,t)={\gamma }_1(t)$
${\alpha }_{21}u(x,t)={\gamma }_2(t)$

Wave Equation with resistive damping

In the homogeneous problem $u_{xx}=\frac{1}{c^2}{u}_{tt}+k{u}_t$