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.

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

Finding a solution: u(x,t)

Let $u(x,t)=X(x)T(t)$
then substitute this into the PDE.
$X''T={\frac {1}{c^{2}}}XT''$
${\frac {X''}{X}}={\frac {1}{c^{2}}}{\frac {T''}{T}}=\mu$
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 $u_{xx}-{\frac {1}{c^{2}}}u_{tt}=0$
$\displaystyle \alpha _{11}u(x,t)=\gamma _{1}(t)$
$\displaystyle \alpha _{21}u(x,t)=\gamma _{2}(t)$

Wave Equation with resistive damping

In the homogeneous problem $u_{xx}={\frac {1}{c^{2}}}u_{tt}+ku_{t}$