# Stationary WavesStationary Waves

A stationary wave forms when two progressive waves with the same frequency travelling in opposite directions are superposed. As they have same frequency, at certain point the are interphase (node) and at other points they are in phase

(antinode). The separation between adjacent nodes (or antinodes) is equal to half the wavelength of the original progressive wave.

Stationary waves can be demonstrated using microwaves (heats up substance at antinodes if the material cannot rotate; reflect with metal plate and check receiver at different points), stretched strings (taut, vibrating string with masses on end) and air columns.

 Progressive wave Stationary wave
 Energy transfer Energy transferred in direction of wave No net energy transfer Wavelength Minimum distance between two adjacent points oscillating in phase (e.g. peaks and troughs) Twice the distance between adjacent nodes (or antinodes) equals the wavelength of the progressive waves that created the stationary wave Phase differences The phase changes across one complete cycle of the wave. All parts of the wave between a pair of nodes are in phase, and on different sides of the node they are in antiphase. Amplitude All parts of the wave have the same amplitude, assuming no energy loss to the surroundings Maximum amplitude occurs at the antinode then drops to zero at the node.

Harmonics:
the fundamental frequency, 0, is the minimum frequency of a stationary wave on a string. The string can form other stationary waves called harmonics at higher frequencies.

1. 20 0             2L
2. 40 20              L
3. 60 30              L
4. 80 40             5L
5. 100 50         4L

The first harmonic is the fundamental mode of vibration or fundamental frequency.

Stationary waves in air columns: sound waves reflected off a surface can form a stationary wave because the original wave and reflected wave travel in opposite directions so superpose. At the closed end of a tube, there is always a node. At the open end, there is always an antinode.

Note that it is not possible to form the second, fourth, sixth… harmonic in a tube open at one end. This is because you need an antinode or a node at both ends of the tube to produce this harmonic, but where one end is open (antinode) and one end is closed (end) this can never occur.

One can measure the speed of sound in air by formation of stationary waves in a resonance tube. By holding a tuning fork above a tube closed at one end, one can form a stationary wave. The air vibrates at the same frequency as the tuning fork. The length of the tube can be changed by raising and lowering it into the water. When the length of the tube above the water equals   λ, the fundamental mode of vibration will be heard. = λ = 4, were is the length of the tube above the water and the frequency of the tuning fork.