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# Stationary waves overview

## Stationary waves

A stationary (or standing) wave is a wave that remains at a fixed position in space (while causing oscillations).

They can only be produced when at least two progressive waves meet. This means that a single wave source cannot produce stationary waves.

When two waves of equal amplitude travelling in opposite directions meet, they "cancel" each other and form a "stationary" group of oscillating particles (like waves on a guitar string).

There is no net transfer of energy or mass by a stationary wave (as the rate of energy transfer by the backward wave is equivalent to that of the forward wave).

The stationary black wave is a result of the intereference of the progressive red and blue waves, which are moving in opposite directions.

## Formation of stationary waves

Stationary waves can form in two situations:

• The medium through which the wave is travelling moves in a direction opposite to the motion of the wave (resulting in zero net movement).

When a large volume of water which flows over a boulder, part of the water gets reflected back. This results in the creation of a stationary wave over the boulder (known as a hydraulic jump).

• Interference between two coherent, progressive waves with the same amplitude travelling in opposite directions.

Stationary waves form on a guitar string as the wave generated by plucking it gets reflected at the fixed ends of the string.

The stationary black wave is a result of the intereference of the progressive red and blue waves, which are moving in opposite directions.

## Formation of stationary waves through wave reflection

A stationary wave can be created when a progressive wave is reflected off a barrier. This creates a wave of equal amplitude that is coherent with the first wave but moving in the opposite direction.

The interference of the incident and reflected wave creates a stationary wave.

At the point of reflection, the wave would have its phase shifted by $\pi$ radians. This means that the reflected wave would always be in anti-phase to the incident wave at the point of reflection, creating a node at the point of reflection.

## Characteristics of stationary waves

The particles in a stationary wave oscillate with an amplitude that is dependent on their position within the wave.

The positions along the wave where the particles oscillate with zero amplitude (i.e. stationary) are known as nodes.

The positions along the wave that the particles oscillate with maximum amplitude are called anti-nodes.

For a stationary wave made up of two waves moving in opposite directions, this amplitude is twice that of the amplitude of an individual component wave.

All particles between two nodes are all in the same phase (i.e. they reach their maximum displacement at the same time). Two particles separated by a node are $\pi$ radians out of phase (i.e. in anti-phase).

Note that each particle in a stationary wave has a different maximum displacement depending on their position within the wave.

The wavelength of a stationary wave is twice the distance between two nodes (or twice the distance between two antinodes). This is similar to the wavelength of a progressive wave being equivalent to the distance covered by the wave in one full cycle.