Equation of Wave with Constant Velocity/Corollary
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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 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:
Proof
We have by hypothesis 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.
$\blacksquare$
Sources
- 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 2$: $(2)$