Equation of Wave with Constant Velocity/Corollary

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

Let $\paren {\map \phi x}_{t \mathop = 0} = \map f x$ be the wave profile of $\phi$.

Then $\phi$ can be expressed using the equation:
 * $\forall x, t \in \R: \map \phi {x, t} = \map f {x + c t}$

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.