# Standing Waves

Mathematics are used to explain standing waves.

## Travelling Waves

Let us consider the following waves
$y_1 = A \cos (wt - bz)$ and $y2 = B \cos (wt - bz)$
Because of the term $wt - bz$, these two waves travel in the same direction.
If we add these two waves, we obtain another travelling wave of the form $y = y_1 + y_2 = (A + B) \cos (wt - bz)$.

## Standing Waves

Let us now consider the following waves
$y_1 = a \cos (wt - bz)$ and $y_2 = a \cos (wt + bz)$
Note that because of the terms $wt - bz$ and $wt + bz$, the two waves travel in opposite directions.
We now add the two waves
$y = y_1 + y_2 = a \cos (wt - bz) + a \cos (wt + bz)$
Expand and simplify
$y = a \cos (wt) \cos(wt) + a \sin (wt) \sin(wt) + a \cos (wt) \cos(wt) - a \sin (wt) \sin(wt)$
$y = 2a \cos (wt) \cos (bz)$
The terms containing the time and distance $\cos(wt)$ and $\cos(bz)$ are separated and therefore the wave obtained is not a travelling one. It is called a standing wave.