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 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:
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:
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
- 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 2$: $(1)$