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 (*x*_{0}, 0). The line through this point parallel to *x* = *ct*

is given by*x* = *x*_{0 } + *ct.* Since our solution is constant along this line, we must have

is given by

But from the initial data,

where *f* is known. So, for any (*x*, *t*),