2.1.4 Analytic solution of the linear advection equation

For initial condition

$$\rho \left( x, t_0 \right) = \rho_0 \left( x \right)$$ (64)

the advection equation (63) has the general solution

$$\tilde{\rho} \left( x, t \right) = \rho_0 \left( x - v_{} \left[ t - t_0 \right] \right) \enspace .$$ (65)

Proof by checking: it fulfils the initial condition and

$$\frac{\partial \tilde{\rho}}{\partial t} + v_{} \frac{\p... ... \frac{\mathrm{d} \rho_0}{\mathrm{d} x} 1 = 0 \enspace .$$ (66)