Stationary waves, also known as standing waves, occur when two waves with the same frequency and amplitude traveling in opposite directions superpose or combine. This phenomenon is based on the principle of superposition, which states that when two or more waves meet at a point in space, their displacements add algebraically.
Let's consider a simple example of a wave traveling along a string. When a wave is incident on an obstacle or a boundary, it gets partially reflected back and partially transmitted through the boundary. The incident and reflected waves can interfere with each other, resulting in a stationary wave pattern.
The incident wave, traveling towards the boundary, carries energy and causes the particles in the medium (the string) to oscillate up and down. When the wave reaches the boundary, it gets partially reflected, which means that some of the energy is sent back along the string in the opposite direction.
Now, let's examine the interference between the incident and reflected waves. Depending on the phase relationship between them, different patterns can emerge.
Displacement Nodes: At certain points along the string, the incident and reflected waves can be perfectly out of phase, meaning they have opposite displacements at those points. When the crest of one wave coincides with the trough of the other wave, they cancel each other out. As a result, the particles at these points remain stationary, and the displacement is zero. These points are called displacement nodes.
Displacement Antinodes: In contrast to the displacement nodes, there are other points on the string where the incident and reflected waves are perfectly in phase. At these points, the crest of one wave coincides with the crest of the other wave, and the troughs also align. This constructive interference leads to a doubling of the amplitude of the individual waves. The particles at these points experience the maximum displacement, resulting in displacement antinodes.
By continuously interfering and superposing, the incident and reflected waves create a stationary wave pattern consisting of alternating nodes and antinodes. The nodes occur at regular intervals, with adjacent nodes separated by half a wavelength. The antinodes are also spaced at half-wavelength intervals, positioned halfway between adjacent nodes.
It is important to note that stationary waves do not propagate or transfer energy in the same way as traveling waves. The energy associated with a stationary wave is localized and remains within the region bounded by the nodes. This is why stationary waves are referred to as "standing" waves since they appear to be fixed in space.
The phenomenon of stationary waves and the formation of nodes and antinodes are not limited to waves on a string but can also occur in other systems, such as sound waves in pipes or electromagnetic waves in resonant cavities. The principle of superposition remains the underlying concept that describes how the interference of incident and reflected waves leads to the formation of these patterns.