Boundary Value Problems/Lesson 6

Return to BVP main page Boundary Value Problems

Derivation of the wave equation using string model.

${\displaystyle \alpha _{11}u(a,t)+\alpha _{12}u_{x}(a,t)=\gamma _{1}}$ ${\displaystyle \alpha _{21}u(b,t)+\alpha _{22}u_{x}(b,t)=\gamma _{2}}$

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

Let ${\displaystyle u(x,t)=X(x)T(t)}$
then substitute this into the PDE.
${\displaystyle X''T={\frac {1}{c^{2}}}XT''}$
${\displaystyle {\frac {X''}{X}}={\frac {1}{c^{2}}}{\frac {T''}{T}}=\mu }$
Where ${\displaystyle \mu }$ is a constant that can be positive, zero or negative. We need to check each case for a solution.

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

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