Equation of Wave with Constant Velocity

Theorem
Let $\phi$ be a wave which is propagated along the $x$-axis in the positive 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
Let us imagine a snapshot of $\phi$ at the time $t = 0$.

Then,, the wave $\phi$ is described by the equation:
 * $\phi = \map f x$

Also, $\phi$ is propagated with no change of shape.

Hence, an imagined snapshot of $\phi$ at the general time $t$ will be:
 * identical with that at $t = 0$
 * moved along the $x$-axis a distance $s$.

By SUVAT:
 * $s = c t$

in the positive direction along the $x$-axis.

Let us set the origin at the point $x = c t$.

Let the distances measured from this new origin be $X$.

Then we have:
 * $X = x + c t$

Hence the new equation for the wave profile of $\phi$ is:
 * $\phi = \map f X$

But referred to that original fixed origin, this means:
 * $\phi = \map f {x - c t}$