Chapter 8. Superposition Flashcards
What is interference?
Interference of waves is the phenomenon when two waves meet or superimpose each other while traveling along the same medium
State the principle of superposition
When two waves are superposed - are in the same place at the same time, the resultant displacement at any point is found by adding the displacements of each separate wave
The two types of interference
Constructive - waves add up
Destructive - waves cancel each other out
Note
Path difference of one wavelength is equal to a phase difference of 360 degrees
path difference of a whole number of wavelengths gives constructive interference
path difference of an extra half wavelength gives a destructive interference
Differentiating progressive waves and stationary waves
Progressive – Stationary
Transfers energy – stores energy
All points on the wave vibrate with the same amplitude – The amplitude of the wave varies from zero on a node to a maximum on an anti-node
Every particle oscillates over the same path, but there is a phase lag between each particle and the one before it – between two nodes, all all of the particles oscillate in phase; on either side of a node they are in anti-phase
The waveform advances with the speed of the wave – The waveshape does not travel along the string
All points within one wavelength have different phases – All points between two adjacent nodes move in phase with one another.Points separated by one node move out of phase by 180 degrees
Define a progressive wave
A wave that generally travels continuously in a medium of the same direction without the change in its amplitude
Define a stationary wave
A wave that oscillates in time but whose peak amplitude profile does not move in space.
A wave formed by the superposition of two progressive waves of the same frequency and amplitude travelling in the opposite direction
What is resonance?
A phenomenon when one object vibrates at the natural frequency of another object causing it to vibrate at a great amplitude
What are harmonics?
Natural frequencies of a system
Why do stationary waves occur at set frequencies only?
This is because there must be a whole number of stationary wave loops fitting into the length of the string
What is the fundamental frequency?
The lowest frequency possible, the first harmonic
Note
speed, c(ms-1) = the square root of (tension, T(N)/mass per unit length, μ(kgm-1))
Note
A closed pipe has one side closed and the other opened and an open pipe has both ends open
A node always forms at the closed end and an anti-node always forms at the opened end
Define diffraction
the process by which a beam of light or other system of waves is spread out as a result of passing through a narrow aperture or across an edge
Note
When waves pass through a gap that is similar in size to their wavelength, there is a lot of diffraction but when the gap is way larger they diffract a little at the edges.
Note
When a wave diffracts, the part where there’s the widest band of the wave is called the central maximum it if followed on both sides by a series of minimums and subsidiary maxima(smaller bands of the wave that get fainter as we go out from the centre)
The maxima occur where light from some parts of the slit superposes constructively with light from other parts of the slit and the minimums occure where there is destructive superposition
Note
sinθ = λ/d
where θ is the angle between the central maximum and the first minimum
λ is the wavelength
and d is the width of the gap(slit) for diffraction
NB: The greater the wavelength of the wave the greater the θ
Note
In a telescope the gap that causes the waves to diffract is the objective lens which is a hole not a slit so the maxima and minima of the diffraction pattern become circles instead of bands. So when viewing stars they came out as discs surrounded by haloes of the subsidiary maxima
Note
According to Rayleigh’s criterion if sinθ is greater than or equal to λ/d then the two sources can be resolved
where θ is the angle between the central maximum and the first minimum
λ is the wavelength
and d is the width of the gap for diffraction
Resolution simply means the two sources can be clearly seen that they are different(they do not overlap much)
Note
For two-source interference, for interference maxima, path difference = nλ where n = 0, 1, 2, 3, …
and for interference minima, path difference = (n + 0.5λ) where n = 0, 1, 2, 3, …
Note
For two-source interference fringes to be observed the two sources have to be coherent(they have the same frequency and maintain the same phase difference), intense and parallel
Note
According to Thomas Young if the distance(a) between two slits diffracting light is much greater than the distance(D) between the slits and a screen then
λ = ax/D
where λ is the wavelength of light
a is the distance between the two slits
x is distance between the centres of two consecutive maxima or minima on the screen
and D is the distance between the slits and the screen
Note
Interference fringes are the series of strips on a screen after two-source interference of light occurs
Note
For a difraction grating
nλ = d sinθn where d is the distance between two slits
and n is the nth-order maximum starting from 1 and going forward
Note
The second harmonic is the first overtone and the third harmonic is the second overtone and so on and so on.
As for pipes if the pipe is a closed pipe(one end open one end closed) then the third harmonic is first overtone and the fifth harmonic is the second overtone as harmonics in closed pipes occur at odd numbers of wavelengths over four.
For open pipes(both ends open) wavelengths occur in normal intgers 1,2,3,.. and the frequency is occurs as halves not quarters. The first overtone is the second harmonic and the second overtone is the third harmonic and so on
An overtone is a multiple of the fundamental frequency.
What is the end correction?
The distance at which the antinode actually is beyond the opening of the pipe. This is because the air just outside the end of the pipe is set into vibration.
Note
For a closed pipe
L1 + c = λ/4 ( c = end correction )
L2 + c = 3λ/4
What is interference?
Interference is the superposition of waves from a finite number of coherent sources resulting in a change of intensity or displacement where the waves meet. Two sources are frequently used.
Coherent sources are those which have a constant phase difference i.e. the sources emit waves of the same wavelength or frequency which are always in phase with each other.
What is path difference?
Path difference is the extra distance which one set of waves travels from the source to the observer, compared with the other set of waves. Path difference is conveniently measured in wavelengths.
Conditions for a steady two-sourced interference pattern to be observed
(i) have the same wavelength or frequency
(ii) have the same amplitude,
(iii) have coherence,
(iv) be un-polarised or both polarised in the same plane.
Note
The separation of the nodal and anti-nodal lines increases as:
-the distance from the two sources increases,
-the smaller the separation of the dippers,
-the wavelength increases.
–nodal line – interference minima
–anti-nodal line – interference maxima
Note
To observe interference bands or fringes, the following conditions are needed:
(i) coherent sources
(ii) a region where two sets of waves can overlap
(iii) the waves must have the same wavelength
(iv) the waves must have similar intensities, so their similar amplitudes can cancel at destructive interference
(v) small path length differences of a few wavelengths.
