Phase Difference
In the above derivation, the concept of repetition of waves was discussed. In order to understand how the motion of two separate points in a wave act in relation with each other, an understanding of Phase Difference is required.
Consider the two waves below :-
Both waves are :-
1. Sinusoidal
2. Identical Wavelength
3. Identical Frequency
4. Identical Amplitude
By comparing how the motion of the medium acts at different parts of the wave, Phase Difference can be explained.
Points 0 and 3 are separated by one Wavelength, therefore perform the identical motion, and are said to have a Phase Difference of 2π radians . These points are in Phase.
Points 0 and 2 are separated by a half wavelength, therefore perform the opposite motion to each other, and are said to have a Phase Difference of π radians. These points are exactly out of Phase.
All important points on a wave are summarised below :-
Also, by comparing the above table, it can be shown that :-
As the Wavelength is unchanging, the above ratio is equal to a constant value.
This implies that if Phase Difference (ϕ) between two points on the wave is equal to x, then :-
By rearranging, the general form of this formula can be found :-
Where :-
ϕ = Phase Difference ( radians )
x = separation of points ( m )
λ = wavelength of the wave ( m )
Example 2 -
A travelling wave has a wavelength of 60mm. A point P is 75mm from the wave's origin and a point Q is 130mm from the wave's origin.
Calculate :-
1. The Phase Difference between P and Q
2. Identify which following Phrase best describes this Phase Difference :-
a. Almost completely out of Phase
b. Almost in Phase
c. Exactly out of Phase
d. Approximately 1/4 cycle out of Phase
Answer :-
Separation of points = 130mm - 75mm = 55mm = 0.055m
Phase Difference = 2π ( 0.055 / 0.060 ) = 5.76 radians
P and Q are separated by 55mm, which is close to one whole wavelength ( 60mm ), meaning that they are almost in Phase.
Phase Change and Reflection
Interference patterns can be seen when a wave is reflected from a surface.
This is because that during reflection, a phase change occurs.
As can be seen from the above diagram, the reflected pulse has a phase difference of π compared to the incident pulse.
There is a similar phase change within light, when incident on a glass surface.
However, there is a slight additional point...
If :-
Moving to a higher optical density (eg. air to glass), a phase change of π occurs.
Moving to a lower optical density (eg. glass to air), no phase change occurs.
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.
Non Examinable - 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 waves must be considered.
Stationary waves are formed by the interference of two waves, of the same frequency and amplitude, travelling in opposite directions.
SQA Exam definition : Stationary waves are formed when transmitted and reflected waves interfere with each other.
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 :-
Non Examinable Derivation - Harmonics
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 stationary 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 stationary 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 stationary 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.
