Equation of Wave with Constant Velocity/Corollary

Theorem
Let $\phi$ be a disturbance which is propagated along the $x$-axis in the negative direction with constant velocity $c$ and without change of shape.

Let $\map f x$ be the wave profile of $\phi$.

Then: $\phi = \map f {x + c t}$ where:
 * $x$ denotes the distance from the origin along the $x$-axis
 * $t$ denotes the time.

Proof
We have that the velocity of $\phi$ in the negative direction is $c$.

Hence the velocity of $\phi$ in the positive direction is $-c$.

By Equation of Wave with Constant Velocity:
 * $\phi = \map f {x - \paren {-c} t}$

Hence the result.