Stationary Waves

Superposition of Waves

Travelling waves can pass by each other without being changed themselves, only the medium is affected. When two travelling waves pass the same point at the same time, the effect they have on the medium depends upon their Phase difference.

If the waves :-

1. Meet in Phase (Peak meets Peak, ϕ = 2π) - Constructive Interference

2. Meet out of Phase (Peak meets Trough, ϕ = π) - Destructive Interference

However, if the waves do not meet at these points, but at some other Phase difference, a combination of effects can be seen.


Periodic waves and Fourier Series

All periodic waves (waves with a repeating pattern) can be mathematically expressed as a combination of sine and cosine waves, which is called a Fourier series.

The Fourier series below can be used to express a "saw tooth" wave :-

The more terms added to the Fourier series, the better the representation of the wave is, however, by looking at only the first four terms for ease of understanding, the following graphs can be generated:-

The above graph shows the first four terms of the above Fourier series plotted onto the same axis. By superimposing these together, a graph of the overall resultant wave can be generated :-

Stationary Waves

So far in this section, the focus has been placed upon travelling waves. In order to fully understand wave behaviour, Stationary or Standing waves must be considered. A Stationary wave occurs when two identical waves interfere with each other, whilst travelling in opposite directions.

Stationary Waves do not travel through a medium, but instead appear to remain in a fixed position, as shown in the diagram below :-

Where :-

A = an Anti-Node (a point of Constructive Interference)

N = a Node (a point of Destructive interference)

The video below shows how Stationary waves can be generated inside a tank of water :-

Harmonics

The below diagram shows a Stationary wave between a fixed "wall" of separation L :-

In order for a Stationary wave to be set up between any two fixed points, fixed nodes must be present at each end. Due to this, only specific multiples Wavelengths can generate a Stationary wave between two fixed points.

Note - for an open ended tube, the standing wave within has anti-nodes at each end. The distance between anti-nodes is the same as the distance between nodes and is equal to λ/2. This means the below derivation applies to all Standing wave systems.

In the above diagram, the longest possible Wavelength is twice the distance between nodes, and therefore the fundamental mode (lowest Frequency) is derived by :-

λ1 = 2L

f1 = v/λ = v/2L

By analysis of the diagram above, it can be shown that :-

λ1 = 2L, λ2 = 2L/2, λ3 = 2L/3 and so on...

This gives a general form of :-

λn = λ1/n

By combining with the earlier calculation of the fundamental Frequency, the allowed frequencies of a standing wave are given by :-

fn = nv/2L = nf1

These different frequencies of possible stationary waves are known as harmonics of the system.

Example 1 -

An organ pipe has a length of 2.00m and is open at both ends. The fundamental Stationary sound wave in the pipe has an anti-node at each end and a node at the centre. The speed of sound in air is equal to 340 ms-1

Calculate :-

1. The Wavelength of the note

2. The Frequency of the note


The separation between two anti-nodes is equal to λ/2, therefore :-

λ/2 = 2.00 m

λ = 4.00 m


The fundamental frequency has a value of n = 1, therefore :-

fn = nv/2L

f = ( 1x340 )/( 2x2.00 )

f = 85.0 Hz


Note - Obviously f could also have been found using v = f λ

The video below shows how Stationary waves acting in multiple directions can create Chladni Patterns on a metal Plate.