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
Non Examinable Derivation - 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 .
