MATH529: L07 Wave Equation

Overview

• Problem statement

• Separation of variables

• General solution

• Boundary value problems:

  • plucked string

  • standing waves

  • superposition

Problem statement

• String of length $L$ with initial shape $f\left(x\right)$, initial velocity $g\left(x\right)$, pinned ends

• Denote vertical displacement at time $t$, position $x$ by $u(x,t)$

• Wave equation

  ${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}=a^2\frac{{\partial }^{2}u}{\partial x^2},a\mathrm{wave}\mathrm{speed}}$

• Initial conditions

  ${\displaystyle u(x,t=0)=f\left(x\right),u_t(x,t=0)=g\left(x\right)}$

• Boundary conditions

  • pinned ends: $u(x=0,t)=0$, $u(x=L,t)=0$, (e.g., $u_0=u_1=0$)

Separation of variables

• $u_{tt}=a^2{u}_{xx}$, $u(x,0)=f\left(x\right)$, $u_t(x,0)=g\left(x\right)$, $u(0,t)=0$, $u(L,t)=0$. (Dirichlet problem)

• $u(x,t)=X\left(x\right)T\left(t\right)\Rightarrow$ $X{T}^{\text{'}\text{'}}=a^2{X}^{\text{'}\text{'}}T$

  ${\displaystyle \frac{1}{a^2}\frac{T^{\text{'}\text{'}}}{T}=\frac{X^{\text{'}\text{'}}}{X}=-\lambda }$

  • $\lambda =0\Rightarrow X^{\text{'}\text{'}}=0\Rightarrow X\left(x\right)=c_1+c_{2}x$, $T^{\text{'}\text{'}}=0\Rightarrow T\left(t\right)=c_3+c_{4}t$.

    $u(x,t)=X\left(x\right)T\left(t\right)=(c_1+c_{2}x)(c_3+c_{4}t)=(1+b_{1}x)(b_2+b_{3}t)$

    $u(0,t)=(b_2+b_{3}t)=0\Rightarrow b_2=b_3=0$$$. Only solution $u(x,t)=0$ $✠$

  • $\lambda =-{\alpha }^2<0\Rightarrow X^{\text{'}\text{'}}-{\alpha }^{2}X=0\Rightarrow X\left(x\right)=c_4\cosh (\alpha x)+c_5\sinh (\alpha x)$

    $T^{\text{'}\text{'}}-a^2{\alpha }^{2}T=0\Rightarrow T\left(t\right)=c_6\cosh (a\alpha t)+c_7\sinh (a\alpha t)$ $✠$

  • $\lambda ={\alpha }^2>0\Rightarrow$$X^{\text{'}\text{'}}+{\alpha }^{2}X=0\Rightarrow X\left(x\right)=c_7\cos (\alpha x)+c_8\sin (\alpha x)$,

    $T^{\text{'}\text{'}}+a^2{\alpha }^{2}T=0\Rightarrow T\left(t\right)=c_9\cos (a\alpha t)+c_{10}\sin (a\alpha t)$

Eigenvalues of the solution

• $u_{tt}=a^2{u}_{xx}$, $u(x,0)=f\left(x\right)$, $u_t(x,0)=g\left(x\right)$, $u(0,t)=0$, $u(L,t)=0$. (Dirichlet problem)

• Only possible choice of $\lambda ={\alpha }^2>0\Rightarrow$

  ${\displaystyle u(x,t)=[b_1\cos (\alpha x)+b_2\sin (\alpha x)][b_3\cos (a\alpha t)+\sin (a\alpha t)]}$

  • $u(0,t)=0\Rightarrow u(0,t)=b_1$$[b_3\cos (a\alpha t)+\sin (a\alpha t)]=0\Rightarrow b_1=0$

  • $u(L,t)=0\Rightarrow$$u(L,t)=b_2\sin (\alpha L)[b_3\cos (a\alpha t)+\sin (a\alpha t)]=0$

    • $b_2=0\Rightarrow u(x,t)=0✠$

    • $\sin (\alpha L)=0\Rightarrow \alpha L=n\pi \Rightarrow {\alpha }_n=\frac{n\pi }{L}$ all lead to a possible solution

      ${\displaystyle u_n(x,t)=\sum _{n=1}^{\infty }\sin ({\alpha }_{n}x)[A_n\cos (a{\alpha }_{n}t)+B_n\sin (a{\alpha }_{n}t)]}$

Fourier sine series solution

• $u(x,t=0)=f\left(x\right)={\sum }_{n=1}^{\infty }{A}_n\sin ({\alpha }_{n}x)$,

  ${\displaystyle A_n=(f\left(x\right),\sin \left({\alpha }_{n}x\right))=\frac{2}{L}\underset{0}{\overset{L}{\int }}f\left(x\right),\sin \left({\alpha }_{n}x\right)dx}$

• $u_t(x,t=0)=g\left(x\right)=$${\sum }_{n=1}^{\infty }a{\alpha }_n{B}_n\sin ({\alpha }_{n}x)$

  ${\displaystyle B_n=(f\left(x\right),\sin \left({\alpha }_{n}x\right))=\frac{2}{a{\alpha }_{n}L}\underset{0}{\overset{L}{\int }}g\left(x\right),\sin \left({\alpha }_{n}x\right)dx}$

• Solution is

  ${\displaystyle u_n(x,t)=\sum _{n=1}^{\infty }\sin ({\alpha }_{n}x)[A_n\cos (a{\alpha }_{n}t)+B_n\sin (a{\alpha }_{n}t)]?}$

Damped wave equation

• Damped harmonic oscillator $y^{\text{'}\text{'}}+f{y}^{\text{'}}+ky=0$

• Damped wave equation $u_{tt}+f{u}_t=a^2{u}_{xx}$

  ${\displaystyle \frac{T^{\text{'}\text{'}}+f{T}^{\text{'}}}{T}=a^2\frac{X^{\text{'}\text{'}}}{X}=-{\alpha }^2\Rightarrow }$ ${\displaystyle X^{\text{'}\text{'}}+\frac{{\alpha }^2}{a^2}X=0}$ ${\displaystyle T^{\text{'}\text{'}}+f{T}^{\text{'}}+{\alpha }^{2}T=0}$