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.



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