Chapter 12: Waves 2

Question 1

Describe what happens when two waves of the same type overlap. How can their instantaneous displacement be found.

Answer 1

Where two waves of the same type overlap, or superpose, they produce a single wave whose instantaneous displacement can be found using the principle of superposition of waves.

Question 2

Describe the principle of superposition.

Answer 2

The principle of superposition states that when two waves meet at a point, the resultant displacement at that point is equal to the sum of the displacements of the individual waves.

Question 3

1. What is constructive interference?
2. How does constructive interference change the intensity?

Answer 3

1. Constructive interference is where two waves are in phase so the maximum displaements (the peaks in a transverse wave) from each wave line up, creating a resultant displacement with increased amplitude.
2. Constructive interference results in an increased amplitude. As intensity ∝ (amplitude)², this increase in amplitude increases the intensity exponentially.

Question 4

1. What is destructive interference?
2. What happens when two waves which have the same amplitude and are in antiphase interfere?

Answer 4

1. Destructive interference is where two waves are in antiphase so the max positive displacement (the peak) lines up with the max negative displacement (the trough) from the other; and the resultant displacement is smaller than for each individual wave.
2. If two waves have the same amplitude and are in antiphase, the waves will interfere destructively and, as a result, the resultant wave will have zero amplitude – it is cancelled out completely.

Question 5

1. What is coherence?
2. What conditions are required for coherence?
3. Why is it not possible to produce stable interference patterns using two filament lamps?

Answer 5

1. Coherence refers to waves emitted from two sources having a constant phase difference.
2. In order to be coherent, the two waves must have the same frequency.
3. It is not possible to produce stable interference patterns using two filament lamps as they do not emit coherent light (different frequencies and changing phase difference).

Question 6

Explain the phenomena of interference patterns containing a series of maxima and minima.

Refer to constructive and destructive interference, and the path difference.

Answer 6

• These maxima and minima are a result of the two waves having travelled different distances from their sources.
  
  • At a max, the waves interfere constructively and at a min, they interfere destructively.
• The difference in distance is called the path difference.
  
  • If the path difference to a point is zero or a whole number of wavelengths (0, 𝛌, 2𝛌, 3𝛌,…) then the two waves will always arrive at that point in phase. This produces constructive interference at that point and is a point of minimum amplitude.
  • If the path difference to a point is an odd number of half wavelengths (1/2𝛌, 3/2𝛌, 5/2𝛌,…) then the two waves will always arrive at that point in antiphase. This produces destructive interference at tat point and is a point of minimum amplitude.

Question 7

Describe the Young double-slit experiment.

What does the Young double-slit experiment demonstrate?

Answer 7

• Two coherent waves are needed to form an interference pattern. The double-slit achieves this by using a monochromatic source of light (which can be achieved using a colour filter that only allows a specific frequency of light to pass) and a narrow single slit to diffract the light.
• Light diffracting from the single slit arrives at the double slit in phase. It then diffracts again at the double slit.
  
  • Each slit acts as a source of coherent waves, which spread from each slit, overlapping and forming an interference pattern that can be seen on a screen as alternating bright and dark regions called fringes.
• The Young double-slit experiment demonstrates the wave nature of light.

Question 8

Describe the derivation of the equation used for calculating 𝛌 from the double-slit experiment.

Answer 8

Question 9

1. What is a stationary wave?
2. How does a stationary wave form?

Answer 9

1. A stationary (or standing) wave is a wave that remains in a constant position with no net transfer of energy.
2. A stationary wave forms when two progressive waves with the same frequency and amplitude travelling in opposite directions are superposed.

Question 10

1. Describe what is meant by a node.
2. Describe what is meant by an antinode.
3. What is the separation between two adjacent nodes?
4. Describe, in terms of energy transferred, how standing waves are different from progressive waves.

Answer 10

1. At points where the progressive waves that make up a standing wave are in antiphase (their displacements cancel out), a node is formed. This is a point on a standing wave that the displacement is always zero.
   The amplitude and intensity is also zero here.
2. At other points where the two waves are always in phase, an antinode is formed. This is the point on a standing wave where the greatest amplitude (and ∴ intensity) is achieved.
3. The separation between two adjacent nodes (or antinodes) is equal to half the wavelength of the original progressive wave.
4. As the two progressive waves are travelling in opposite direction, there is no net energy transfer by a stationary wave, unlike a single progressive wave.

Question 11

1. Describe the displacement of particles in between adjacent nodes.
2. Describe the displacement of particles on different sides of a node.

Answer 11

1. In between adjacent nodes, all the particles in a stationary wave are oscillating in phase with each other. Hence, they all reach their max positive displacement at the same time, with the max ampltiude at the antinode.
2. On different sides of a node, the particles are in antiphase, with a phase difference of π radians. The particles on one side of a node reach their mac positive displacement at the same time as those on the other side reach their max negative displacement.

Question 12

1. What is the fundamental frequency?
2. What is the fundamental mode of vibration?

Answer 12

1. The fundamental frequency is the lowest frequency at which an object (e.g. an air column in a pipe or a string fixed at both ends) can vibrate.
2. The fundamental mode of vibration refers to a vibration at the fundamental frequency.

Question 13

1. Describe how a stationary wave can be formed on a string.
2. When the string is plucked, it vibrates in its fundamental mode of vibration. Express the wavelength of the wave formed in terms of the length of the string L.

Answer 13

1. If a string is stretched between two fixed points, these points act as nodes. When the string is plucked, a progressive wave travels along the string and reflects off its ends. This creates two progressie waves travelling in opposite directions: a stationary wave.
2. In the fundamental mode of vibration, the wavelength of the progressive wave is equal to 2L.

Question 14

1. What are harmonics on a string? When are they formed?
2. Deduce the relationship between frequency and wavelength for harmonics on a string.

Answer 14

1. Along with the fundamental mode of vibration, a string stretched between two fixed points can form other stationary waves called harmonics at higher frequencies.
2. For a given string of fixed tension, the speed of the progressive waves along the string is constant. From the equation for progressive waves, v = f 𝛌, you can see that as the frequency increases, the wavelength must decrease in proportion.

Answer 15

Question 16

1. Describe the principle behind making stationary sound waves.
2. Describe how stationary sound waves can be made in tubes.

Answer 16

1. Sound waves reflected off a surface can form a stationary wave. The original wave and the reflected wave travel in opposite directions and superpose.
2. Stationary sound waves can be made in tubes by making the air column inside the tube vibrate at frequencies related to the length of the tube. The stationary wave formed depends on whether the ends of the tube are open or closed.

Question 17

Describe and explain how a stationary wave is formed in a tube closed at one end

Answer 17

• In order for a stationary wave to form in a tube closed at one end, there must be an antinode at the open end and a node at the closed end.
  
  • The air at the closed end cannot move, and so must form a node.
  • At the open end, the oscillations of the air are at their greatest amplitude, so it must be an antinode.
• The fundamental mode of vibration simply has a node at the base and an antinode at the open end. Harmonics are also possible.
  
  • Unlike stationary waves on stretched strings, in a tube closed at one end, it is not possible to form a harmonic at 2f₀, 4f₀ etc.
  • The frequencies of the harmonics in tubes closed at one end are always an odd multiple of the fundamental frequency (3f₀, 5f₀ etc.)

Question 18

Describe how a stationary wave can be formed in an open tube.

Answer 18

• A tube open at both ends must have a antinode at each end in order to form a stationary wave.
• Unlike a tube closed at one end, harmonics at all integer multiples of the fundamental frequency are possible in an opne tube (f₀, 2f₀, 3f₀,…)