Chapter 8. Superposition

Question 1

What is interference?

Answer 1

Interference of waves is the phenomenon when two waves meet or superimpose each other while traveling along the same medium

Question 2

State the principle of superposition

Answer 2

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

Question 3

The two types of interference

Answer 3

Constructive - waves add up
Destructive - waves cancel each other out

Question 4

Note

Answer 4

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

Question 5

Differentiating progressive waves and stationary waves

Answer 5

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

Question 6

Define a progressive wave

Answer 6

A wave that generally travels continuously in a medium of the same direction without the change in its amplitude

Question 7

Define a stationary wave

Answer 7

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

Question 8

What is resonance?

Answer 8

A phenomenon when one object vibrates at the natural frequency of another object causing it to vibrate at a great amplitude

Question 9

What are harmonics?

Answer 9

Natural frequencies of a system

Question 10

Why do stationary waves occur at set frequencies only?

Answer 10

This is because there must be a whole number of stationary wave loops fitting into the length of the string

Question 11

What is the fundamental frequency?

Answer 11

The lowest frequency possible, the first harmonic

Question 12

Note

Answer 12

speed, c(ms^-1) = the square root of (tension, T(N)/mass per unit length, μ(kgm^-1))

Question 13

Note

Answer 13

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

Question 14

Define diffraction

Answer 14

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

Question 15

Note

Answer 15

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.

Question 16

Note

Answer 16

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

Question 17

Note

Answer 17

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

Question 18

Note

Answer 18

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

Question 19

Note

Answer 19

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)

Question 20

Note

Answer 20

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, …

Question 21

Note

Answer 21

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

Question 22

Note

Answer 22

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

Question 23

Note

Answer 23

Interference fringes are the series of strips on a screen after two-source interference of light occurs

Question 24

Note

Answer 24

For a difraction grating
nλ = d sinθ_n where d is the distance between two slits
and n is the nth-order maximum starting from 1 and going forward

Question 25

Note

Answer 25

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.

Question 26

What is the end correction?

Answer 26

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.

Question 27

Note

Answer 27

For a closed pipe
L₁ + c = λ/4 ( c = end correction )
L₂ + c = 3λ/4

Question 28

What is interference?

Answer 28

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.

Question 29

What is path difference?

Answer 29

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.

Question 30

Conditions for a steady two-sourced interference pattern to be observed

Answer 30

(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.

Question 31

Note

Answer 31

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

Question 32

Note

Answer 32

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.