4.4.4 stationary waves

Stationary waves are also known as standing waves. They can be created by longitudinal and transverse waves and form when two progressive waves with the same frequency (and ideally the same amplitude) travelling in opposite directions superpose. At points in antiphase the waves cancel out forming a node. At points in phase the waves cancel out forming an antinode. Where displacement is zero, amplitude and therefore intensity are also zero. The separation between adjacent nodes (or antinodes, for that matter) equates to half the wavelength of the original progressive waves. It is important to realise that there is no net transfer of energy as the two progressive waves are travelling in opposite directions so they sort of cancel each other out.

All the particles between adjacent nodes are oscillating in phase with one another. This is because, although they have different amplitude, they all reach their maximum positive displacements at the same time. On different sides of a node the particles are in antiphase; the particles to the left of a node reach their maximum positive displacement at the same time the particles on the right reach their maximum negative displacement.

This set of graphs nicely demonstrates the motion of a stationary wave:

We need to know how to demonstrate stationary waves using microwaves, stretched strings, and air columns:

Microwaves
We can form a stationary wave by reflecting microwaves off a metal sheet so that two microwaves of the same frequency are travelling in opposite directions. Using a microwave receiver we can detect the changes in intensity between nodes (low/no intensity) and antinodes (maximum intensity). The distance between the transmitter and the metal sheet must be adjusted until the receiver detects a series of notes/antinodes. As we are aware already, the distance between adjacent nodes or antinodes equates to half the wavelength of the microwaves from the transmitter.

Stretched strings
Each string has a fundamental mode of vibration. The frequency of this vibration is the fundamental frequency (f₀). This depends on factors such as the strings mass, tension, and length. When a string is stretched between two points the two points act as nodes. If the string is plucked a progressive wave travels along the string and reflects off its ends which creates two progressive waves travelling in opposite directions and these superpose and a stationary wave is formed. When plucked, the string vibrates in its fundamental mode of frequency, the wavelength of the progressive wave is double the length of the string.

The fundamental frequency (f₀) is the minimum frequency of a stationary wave for a string. However, it is possible to form other stationary waves known as harmonics at higher frequencies. For a given string at a fixed tension the speed of the progressive wave is constant. From v = fλ we can see that as frequency increases λ decreases proportionally. E.d at a frequency of 2f₀ the wavelength is half what it was at f₀. This table (from the kerboodle OCR A Physics textbook) nicely demonstrates this:

Air columns
Most woodwind instruments (sound is longitudinal) produce notes from blowing over the top of a tube creating a standing wave inside. This produces a note at a particular frequency (the length of the tube determines the wavelength of the note it produces). Sound waves reflected off a surface can produce a stationary wave. The original wave and the reflected wave travel in opposite directions and superpose. Stationary sound waves can also be made in tubes by making the air column inside the tube vibrate at frequencies related to the length of the tube. The stationary wave formed depends on whether the ends of the tube are open or closed.

Closed at one end:
In order for a stationary wave to form in a tube closed at one end there must be an antinode at the open end and a node at the closed end. The air at the closed end cannot move so it must form a node whilst at the open end the oscillations of the air are at their greatest amplitude so it must be an antinode. The fundamental mode of vibration has a node at the base and antinode at the top, the wavelength is 4 times the length of the tube.

In a tube closed at one end it is not possible to form harmonics at 2f₀, 4f₀, 6f₀, etc. This is because the open end must be an antinode. The frequencies of the harmonics in tubes closed at one end are always an odd multiple of f₀, 3f₀, 5f₀, 7f₀, etc as demonstrated in this diagram:

Open at both ends:
A tube open at both ends will have an antinode at both ends and a node in the centre (if vibrating at f₀). Harmonics at all integer multiples of the fundamental frequency are possible. This diagram nicely shows this: