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