Equation of Harmonic Wave

Theorem
Let $\phi$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.

Then the displacement of $\phi$ at point $x$ and time $t$ can be expressed using the equation:
 * $\map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$

$\phi$ can also be expressed in the following forms:

Proof
From Wave Profile of Harmonic Wave:


 * $\forall x \in \R: \paren {\map \phi x}_{t \mathop = 0} = a \cos \omega x$

From Equation of Wave with Constant Velocity:
 * $\forall x, t \in \R: \map \phi {x, t} = a \map \cos {\omega \paren {x - c t} }$

Hence the result.