2.1.1 Left end fixed, right end free. Initial velocity zero. Damping present.

Solve, \(0<x<L\)\[ \frac{\partial ^{2}u}{\partial t^{2}}+b\frac{\partial u}{\partial t}=c^{2}\frac{\partial ^{2}u}{\partial x^{2}}\qquad 0<x<L,t>0 \] Boundary conditions, \(t>0\)\begin{align*} u\left ( 0,t\right ) & =0\\ \left . \frac{\partial u}{\partial x}\right \vert _{x=L} & =0 \end{align*}

Initial conditions, \(t=0\)\begin{align*} \frac{\partial u\left ( x,0\right ) }{\partial t} & =0\\ u\left ( x,0\right ) & =f\left ( x\right ) =\left \{ \begin{array} [c]{ccc}\frac{3h}{L}x & & 0<x<\frac{L}{3}\\ h & & \frac{L}{3}<x<L \end{array} \right . \end{align*}

Use \(c=4,h=0.1,L=1.\) Consider three cases for damping, \(b=0.5\frac{\pi c}{L},b=\frac{\pi c}{L},b=2\frac{\pi c}{L}\)

Solution case (underdamped) \(b=0.5\frac{\pi c}{L}\). \begin{align*} \lambda _{n} & =\frac{\left ( 2n-1\right ) \pi }{2L}\\ \omega _{n} & =\lambda _{n}c\\ \xi _{n} & =\frac{b}{2\omega _{n}}\\ C_{n} & =\frac{24h}{\left ( \left ( 2n-1\right ) \pi \right ) ^{2}}\sin \left ( \frac{\left ( 2n-1\right ) }{6}\pi \right ) \\ \beta _{n} & =\omega _{n}\sqrt{1-\xi _{n}^{2}} \end{align*}

\[ u\left ( x,t\right ) =\sum _{n=1}^{\infty }C_{n}e^{-\frac{b}{2}t}\left ( \cos \left ( \beta _{n}t\right ) +\frac{b}{2\beta _{n}}\sin \left ( \beta _{n}t\right ) \right ) \sin \left ( \lambda _{n}x\right ) \]

Solution case (critical damped)

\(b=\frac{\pi c}{L}\).

\[ u\left ( x,t\right ) =C_{1}\left ( e^{\frac{-b}{2}t}+\frac{b}{2}te^{\frac{-b}{2}t}\right ) \sin \left ( \lambda _{1}x\right ) +\sum _{n=2}^{\infty }C_{n}e^{-\frac{b}{2}t}\left ( \cos \left ( \beta _{n}t\right ) +\frac{b}{2\beta _{n}}\sin \left ( \beta _{n}t\right ) \right ) \sin \left ( \lambda _{n}x\right ) \]

Solution case (overdamped damped)

\(b=2\frac{\pi c}{L}\). First mode only overdamped, rest underdamped.\begin{align*} \omega _{1} & =2\frac{\pi c}{L}\\ C_{1} & =\frac{1}{1-\frac{-\frac{b}{2}+\omega _{1}\sqrt{\zeta ^{2}-1}}{-\frac{b}{2}-\omega _{1}\sqrt{\zeta ^{2}-1}}}\frac{24h}{\pi ^{2}}\sin \left ( \frac{\pi }{6}\right ) \\ C_{n} & =\frac{24h}{\left ( \left ( 2n-1\right ) \pi \right ) ^{2}}\sin \left ( \frac{\left ( 2n-1\right ) }{6}\pi \right ) \qquad n>1 \end{align*}

\begin{align*} u\left ( x,t\right ) & =C_{1}\left ( e^{\left ( -\frac{b}{2}+\omega _{1}\sqrt{\zeta ^{2}-1}\right ) t}-\frac{-\frac{b}{2}+\omega _{1}\sqrt{\zeta ^{2}-1}}{-\frac{b}{2}-\omega _{1}\sqrt{\zeta ^{2}-1}}e^{\left ( -\frac{b}{2}-\omega _{1}\sqrt{\zeta ^{2}-1}\right ) t}\right ) \sin \left ( \lambda _{1}x\right ) \\ & +\sum _{n=2}^{\infty }C_{n}e^{-\frac{b}{2}t}\left ( \cos \left ( \beta _{n}t\right ) +\frac{b}{2\beta _{n}}\sin \left ( \beta _{n}t\right ) \right ) \sin \left ( \lambda _{n}x\right ) \end{align*}

Source code for the above 3 cases is