2.3.1 Initial velocity zero case

   2.3.1.1 Animations

Solve the wave equation \(u_{tt}=a^{2}u_{xx}\) for infinite domain \(-\infty <x<\infty \) with initial position \(u\left ( x,0\right ) =f\left ( x\right ) \) and zero initial velocity \(g\left ( x\right ) =0\).  

solution

The solution for wave PDE \(u_{tt}=a^{2}u_{xx}\) on infinite domain can be given as either series solution, or using D’Alembert solution. Using D’Alembert, the solution is\[ u\left ( x,t\right ) =\frac{1}{2}\left ( f\left ( x-at\right ) +f\left ( x+at\right ) \right ) +\frac{1}{2a}\int _{x-at}^{x+at}g\left ( s\right ) ds \] Since \(g\left ( x\right ) \) is zero the above simplifies to\[ u\left ( x,t\right ) =\frac{1}{2}\left ( f\left ( x-at\right ) +f\left ( x+at\right ) \right ) \]

2.3.1.1 Animations

Example 1 Problem Solve the wave equation \(u_{tt}=a^{2}u_{xx}\) for infinite domain \(-\infty <x<\infty \) with initial position \(u\left ( x,0\right ) =f\left ( x\right ) =\frac{1}{1+x^{2}}\) and zero initial velocity \(g\left ( x\right ) =0\).  Let \(a=1\).

solution

The solution for wave PDE \(u_{tt}=a^{2}u_{xx}\) on infinite domain can be given as either series solution, or using D’Alembert solution. Using D’Alembert, the solution is\[ u\left ( x,t\right ) =\frac{1}{2}\left ( f\left ( x-at\right ) +f\left ( x+at\right ) \right ) +\frac{1}{2a}\int _{x-at}^{x+at}g\left ( s\right ) ds \] But here \(a=1\) and \(g\left ( x\right ) \) is zero. Therefore the above simplifies to\[ u\left ( x,t\right ) =\frac{1}{2}\left ( f\left ( x-t\right ) +f\left ( x+t\right ) \right ) \] Where \(f\left ( x-t\right ) \) is the initial position shifted to the right by \(t\) and \(f\left ( x+t\right ) \) is the initial position shifted to the left by \(t\). Since \(f\left ( x\right ) =\frac{1}{1+x^{2}}\), the above becomes\[ u\left ( x,t\right ) =\frac{1}{2}\left ( \frac{1}{1+\left ( x-t\right ) ^{2}}+\frac{1}{1+\left ( x+t\right ) ^{2}}\right ) \] Here is animation for 45 seconds.

Source code for the above animation is

Example 2 Problem Solve the wave equation \(u_{tt}=a^{2}u_{xx}\) for infinite domain \(-\infty <x<\infty \) with initial position \(u\left ( x,0\right ) =f\left ( x\right ) =\sin \left ( x\right ) \) from \(-\pi <x<\pi \) and zero everywhere. And with zero initial velocity \(g\left ( x\right ) =0\).  Let \(a=1\).

solution

The solution for wave PDE \(u_{tt}=u_{xx}\) on infinite domain using D’Alembert solution with zero initial velocity is\begin{align*} u\left ( x,t\right ) & =\frac{1}{2}\left ( f\left ( x-t\right ) +f\left ( x+t\right ) \right ) \\ & =\frac{1}{2}\left ( \sin \left ( x-t\right ) +\sin \left ( x+t\right ) \right ) \end{align*}

The following is an animation Here is animation for 10 seconds.

Source code for the above animation is