Waves Recap
"Waves transfer energy from one point to another with no net transfer of mass"
This means that the particles of the medium do not travel with the wave, apart from being disturbed from their equilibrium position as the wave passes.
As can be seen from the previous section, the formula to describe a wave's displacement is given by :-
y = 0 at t = 0 y = A Sin wt
y = A at t = 0 y = A Cos wt
The Travelling Wave Equation
The diagram below shows a sinusoidal wave ( y = 0 at t = 0 ) :-
In the above diagram, two points on the wave have been marked :-
i ) - displacement on x axis = 0
ii ) - displacement on x axis = x
Given the above points and by applying s = vxt it can be shown that :-
t = x/v
Where :-
t = time taken ( s )
x = displacement from origin ( m )
V = velocity of the wave ( ms-1 )
Since wave motion is repetitive after each Wavelength, the formula describing the wave can be derived as follows :-
At point i ) x = 0, t = x/v
At point ii ) x = x, t = t
The motion is repetitive, therefore motion of particles at i ) and ii ) are identical, allowing the replacement of t within the calculation by :-
t = t - x/v
The general formula can then be derived as follows :
y = a sin ωt
Substitute in t = t - x/v
y = a sin ω(t - x/v)
Substitute in ω = 2 π f and v = f λ
y = a sin 2 π (ft - x/λ)
Note - As sine is involved, ensure that the calculation is performed in radians.
Also, the above derivation applies to moving to the right along the x axis, for a wave moving to the left (-x) direction :-
y = a sin 2 π (ft + x/λ)
The above formula allows the calculation of the Displacement of the medium at any given point or time as long as the Wavelength and Frequency of the wave is known.
Example 1 -
A periodic wave travelling in the x-direction is described by the following equation :-
y = 0.2 sin ( 4π t + 0.1x)
Calculate :-
1. The Amplitude of the wave
2. The Frequency of the wave
3. The Wavelength of the wave
4. The Speed of the wave
In order to complete this question, the above formula must be rearranged to appear in the form of the general equation above :-
y = 0.2 sin ( 4π t + 0.1x )
y = 0.2 sin 2 π [ 2t + (0.1x / 2π) ]
By comparing the two equations together, the following answers can be found :-
a = 0.2 m
f = 2 Hz
λ = 2π /0.1 = 63m
v = 2 x 63 = 126 ms-1
Energy of a Wave
As discussed in Higher Physics, the intensity of a wave is defined as "the amount of energy that passes though unit area perpendicular to the wave direction in unit time".
As also discussed previously, the amplitude and the energy inherent to a wave are related to each other. They are not related proportionally, however, as the amplitude of a wave varies in a sinusoidal manner with time whereas the energy of the wave varies as sine squared.
The graph below shows how the energy of a wave and the amplitude of a wave vary with time:-
Since the variation in amplitude is simple harmonic, the energy variation with time is proportional to sine squared or more commonly :-
"The intensity I (or energy E) of a wave is proportional to the square of its amplitude"
Non-examinable derivation of above :
A SHM wave is a displacement that is resisted by a restoring force. The larger the displacement x, the larger the force F = kx needed to create it. Because work W is related to force multiplied by distance (Fx) and energy is put into the wave by the work done to create it, the energy in a wave is related to amplitude. A wave’s energy is therefore directly proportional to its amplitude squared because W ∝ Fx = kx2 .
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.
