The Transport Equation and Directional Derivatives - Solution of the Transport Equation
Using the gradient operator
we may rewrite equation (1) as
This equation says that the directional derivative in the (1, c) direction (in the t, x-plane) is zero. So our solution u(x, t) must be constant in this direction. In the t, x-plane, the (1, c) direction is along lines parallel to x = ct, which are called the characteristics of equation (1).
Now, fix a point on the x-axis, say (x0, 0). The line through this point parallel to x = ct
is given by x = x0 + ct. Since our solution is constant along this line, we must have