Sum of Equal and Opposite Harmonic Waves form Stationary Wave

Theorem
Let $\phi_1$ be a harmonic wave which is propagated along the $x$-axis in the positive direction with constant velocity $c$.

Let $\phi_2$ be a harmonic wave travelling with constant velocity $-c$, that is, at the same speed as $\phi_1$ but in the opposite direction.

Then their sum $\phi_1 + \phi_2$ describes a stationary wave.

Proof
From Equation of Harmonic Wave: Wave Number and Frequency:
 * $(1): \quad \map {\phi_1} {x, t} = a \map \cos {2 \pi \paren {k x - \nu t} }$

where:
 * $k$ denotes the wave number of $\phi_1$
 * $\nu$ denotes the frequency of $\phi_1$.

From Equation of Wave with Constant Velocity: Corollary, the equation of $\phi_2$ is:
 * $(2): \quad \map {\phi_2} {x, t} = a \map \cos {2 \pi \paren {k x + \nu t} }$

Then we have: