Standing Waves

Mathematics is used to explain standing waves and wave superposition. Explore how combining waves traveling in identical versus opposite directions generates unique wave patterns.

Travelling Waves

Let us consider the following waves:

\[ y_1 = A \cos(\omega t - \beta z) \] \[ y_2 = B \cos(\omega t - \beta z) \]

Because of the term \(\omega t - \beta z\), these two waves travel in the same direction.

If we add these two waves, we obtain another travelling wave of the form:

\[ y = y_1 + y_2 = (A + B) \cos(\omega t - \beta z) \]

Standing Waves

Let us now consider two waves traveling in opposite directions:

\[ y_1 = a \cos(\omega t - \beta z) \] \[ y_2 = a \cos(\omega t + \beta z) \]

Note that because of the terms \(\omega t - \beta z\) and \(\omega t + \beta z\), the two waves travel in opposite directions.

We now add the two waves:

\[ y = y_1 + y_2 = a \cos(\omega t - \beta z) + a \cos(\omega t + \beta z) \]

Using trigonometric expansion identities, we expand and simplify:

\[ y = a \cos(\omega t)\cos(\beta z) + a \sin(\omega t)\sin(\beta z) + a \cos(\omega t)\cos(\beta z) - a \sin(\omega t)\sin(\beta z) \] \[ y = 2a \cos(\omega t) \cos(\beta z) \]

The time-dependent and distance-dependent terms \(\cos(\omega t)\) and \(\cos(\beta z)\) are separated. Therefore, the resulting wave is not a travelling wave; instead, it is called a standing wave.

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