Equation of Wave with Constant Velocity

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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 the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:

$\map \phi {x, t} = \map f {x - c t}$

where:

$x$ denotes the distance from the origin along the $x$-axis
$t$ denotes the time.


Corollary

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 the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:

$\map \phi {x, t} = \map f {x + c t}$


Proof

Let us imagine a snapshot of $\phi$ at the time $t = 0$.

Then, by hypothesis, the wave $\phi$ is described by the equation:

$\phi = \map f x$

Also by hypothesis, $\phi$ 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:

$\phi = \map f {x - c t}$

$\blacksquare$


Sources