Waves & Optics - Superposition

Question 1

Superposition

Answer 1

• Principle of Superposition: “When two or more waves cross at a point, the displacement at that point is equal to the sum of the displacements of the individual waves.”
• The individual wave displacements may be positive or negative. If the displacements are vectors, then the sum is calculated by vector addition.
• Superposition can be used to explain interference, diffraction and standing waves
• Superposition works for any type of wave (sound waves, water surface waves, electromagnetic waves…) but only under certain conditions.
• When waves pass beyond a point of intersection, they separate out again and are unaffected.

Question 2

Conditions for Superposition

Answer 2

• The principle of superposition can be applied to any type of wave providing that:
‣ The waves being superposed are of the
same type
‣ The medium that the waves are
propagating through behaves linearly
• If the waves are also coherent then the superposition resembles another wave with the same frequency

Question 3

Coherence

Answer 3

• If waves from two sources spread out and cross throughout a region of space then superposition occurs at every point, and a pattern of interference is obtained
• The superposition of waves from two sources usually only results in a stationary interference pattern if the sources are coherent
• This means that the waves from the sources have both the same frequency and phase difference

Question 4

Constructive and Destructive interference

Answer 4

• When two coherent waves are superimposed, the amp[litude of the resultant displacement depends on the phase difference of the two waves.
• This can result in three possible outcomes, constructive, destructive, and partial constructive.
• When the waves a phase difference which are equal, the two waves will constructively interfere, this means that the resultant amplitude will be equal to the sum of the two amplitudes
• When the two waves have a phase difference of 180 degrees, they will destructively interfere, this means that the resultant wave will have a amplitude equal to the difference between the two waves.

Question 5

Path and Phase Difference

Answer 5

• Phase is an angle, which gives the fraction of the progress through a repetitive cycle in terms of degrees between 0 and 2π radians or 0 and 360∘
• When two coherent waves are superposed, the resultant wave depends on the phase difference between the two.
• If the waves have followed two different paths to the same point, the phase difference is proportional to the difference in length between the paths, known as the path difference.
• IF the path difference is an integer multiple of the wavelength then the waves are in phase and interfere constructively.
• IF the path difference is a half-integer multiple of the wavelength, then the waves are π rad out of phase and interfere destructively if the sources are in phase.

Question 6

Amplitude and Intensity

Answer 6

• When superposition occurs, for slow mechanical waves the amplitude can usually be observed easily.
• For high-frequency waves, such as sound or EM waves, the amplitude can be calculated by measuring the wave’s intensity.
• Intensity is a measure of the energy of the wave in the region, which is proportional to the square of the amplitude

Question 7

Beats

Answer 7

• When two waves of slightly differing frequencies are superposed at some location, the resultant amplitude appears to vary periodically.
• The phenomenon is known as beating and is most commonly observed with sound waves.