Interference

It is the combination of two or more waves that are in the same region of space generating a resultant wave.

On the left part of the figure we have the sum of two waves (red + blue) completely out of phase, which generates a completely destructive interference and the resulting wave (green) is zero. On the right part, we have the sum of two waves (red + blue) in phase, which generates a constructive interference.

Standing Waves

A stationary wave occurs when two or more waves, in opposite directions, form a single wave. In this wave, certain points do not move. The word "stationary" refers to these points and not to the wave as a whole, which vibrates normally out of these points. The distance between two consecutive nodes is \( \frac{\lambda}{2} \).

This phenomenon can be observed in a rope where one of its ends is tied and the other vibrates. After traveling down the rope, the pulse is reflected back when it reaches the fixed end. The sum of the wave sent with the reflected one can generate a standing wave, depending on the characteristics of the incident wave. See the figure.

Illustration of standing waves. The points of maximum and minimum oscillate continuously between themselves. This region is known as antinode. Two neighboring antinodes are always in opposing phases. The intermediate points are the nodes and do not move. Depending on the speed \(v\), the length \(L\) of the rope and the frequency \(f\), the wave will have different harmonics ( \(n\) ), or a different number of nodes and antinodes at different parameters.

Standing Waves in Tubes

In tubes where one end may or may not be closed, where a sound wave is applied (pressure wave in air), it is also possible to obtain standing wave so that:

Open tubes
\( f_n = \frac{nv}{2L} \)
Closed tubes
\( f_n = \frac{nv}{4L} \)

Beats

The beat is the phenomenon that results from the superposition of two waves of slightly different frequencies. The beat is a variable amplitude perturbation whose frequency is equal to the difference between the frequencies of the two waves; that is, the number of beats per second is equal to the difference between the frequencies of the component waves: \( f_{beat} = ~ ∣f_1 - f_2∣.\)

Illustration of beats. The red and blue waves have very similar frequencies, imperceptible in the image. However, when we add the two waves, the difference is clear. The wave starts to have a beat.