The 1-D wave equation [math]u_{tt}=c^2u_{xx}[/math] for infinite boundaries has the D'Alembert solution through a coordinate transformation [math]u\left(x,t\right)=F\left(x-ct\right)+G\left(x+ct\right)[/math] where [math]F\left(x\right)=\frac{1}{2}f\left(x\right)-\frac{1}{2c}\int_a^xg\left(\xi\right)d\xi[/math], [math]G\left(x\right)=\frac{1}{2}f\left(x\right)+\frac{1}{2c}\int_a^xg\left(\xi\right)d\xi[/math]. The initial conditions are [math]f\left(x\right)=u\left(x,0\right)[/math] and [math]g\left(x\right)=u_t\left(x,0\right)[/math].[br]For the semi-infinite domain, [math]x>0[/math], [math]f\left(x\right)[/math] would be undefined for [math]x<0[/math]. Therefore a boundary condition at [math]x=0[/math] is used to extend to negative values [math]x[/math]. [br]Therefore a boundary condition at [math]x=0[/math] is used to extend to negative values. Then [math]F\left(x\right)=F\left(x\right)\text{ if }x>0\text{ and }-G\left(-x\right)\text{ if }x<0[/math]. Also if [math]u\left(0,t\right)=h\left(t\right)[/math] this boundary condition results in the modification for [math]x<0[/math] [math]F\left(x\right)=-G\left(-x\right)+h\left(\frac{-x}{c}\right)[/math].[br]This applet allows you to input a function for [math]u\left(x,0\right)[/math] and [math]u\left(0,t\right)=h\left(t\right)[/math]. Note use x for t in this term. You can also modify the wave speed, [math]c[/math] and then advance the time, [math]t[/math]

The boundary condition [math]u_{x\left(0,t\right)}=0[/math] can also be accommodated with an extension for [math]F\left(x\right)[/math] to negative [math]x[/math] values. This applet shows the result for [math]u_{x\left(0,t\right)}=0[/math][br]

What was the extension on [math]F\left(x\right)[/math] ?