1D Linear Advection

Time =

Sine Wave

Initial condition: sine function

Here we generate a wave given the initial condition $$\phi(x,0)=f(x)=A\sin(kx)$$ where $$A$$ is the amplitude of the wave and $$k$$ is the wavenumber of the wave.

The speed of the wave $$c$$ is dictated by $$\frac{dx}{dt}=c$$ which gives rise to straight lines called characteristic lines by $$x-ct=a$$ where $$a$$ is a constant. On these lines, $$\phi$$ is conserved and moves with speed $$c$$.

To see this, we fix values $$A$$ and $$k$$ but vary $$c$$. Click "Initial Wave" to see the characteristic lines in the upper panel and the initial wave in the lower panel. Click "Generate Wave" to see the movement of the initial point on the wave in the $$(x,t)$$ plane concomitant with the generated wave in the lower panel.