The Convection Equation is [math]u_t=-Vu_x[/math] where [math]V[/math] is a constant.[br]The solution with an initial condition of [math]u\left(x,0\right)=sin\left(\omega x\right)[/math] is shown along with [math]u+u_tdt=u-Vu_xdt[/math] where [math]dt[/math] is a small time step. This illustrates the direction of change as the red [math]u\left(x,t\right)[/math] curve will move towards the green dotted curve. In this case the short time [math]dt=0.1[/math] for illustration purposes.

Vary the sliders and click the play button. Note what the red curve appears to do.[br][br]How does changing the sliders affect the curve motion?[br][br]How do you translate a general function?[br][br][br]A general function, [math]f\left(x\right)[/math] can be translated along the [math]x[/math]-axis a distance [math]b[/math] by evaluating [math]f\left(x-b\right)[/math] . Note at [math]x=b[/math] the original function parameter would be zero.[br][br]What would this suggest as a possible solution to this problem?[br][br]What happens if you substitute [math]u=sin\left(\omega\left(x-Vt\right)\right)[/math] into the partial differential equation? Note: [math]u_t=-\omega V\cos\left(\omega\left(x-Vt\right)\right)\text{ and }u_x=\omega\cos\left(\omega\left(x-Vt\right)\right)[/math] by the chain rule.