Superposition of Waves (SL IB Physics)

• The principle of superposition states that:

When two or more waves meet, the resultant displacement is the vector sum of the displacements of the individual waves 

• The principle of superposition applies to both transverse and longitudinal waves
• Interference occurs whenever two or more waves superpose

• For a clear stationary interference pattern, the waves must be of the same:
  • Type
  • Amplitude
  • Frequency

• They must also have a constant phase difference

Constructive & Destructive Interference

• Constructive interference occurs when
  • Two or more waves superpose and have displacements in the same direction (both positive or both negative)

• Destructive interference occurs when
  • Two or more waves superimpose and have displacements in opposite directions (one positive and one negative)

• When two waves with the same amplitude meet at a point, they can:
  • Be in phase and interfere constructively, so that the displacement of the resultant wave is double the displacement of each individual wave
  • Be in anti-phase and interfere destructively, so that the displacement of the resultant wave is equal to zero

Waves in superposition can undergo constructive or destructive interference

• Superposition occurs for any two waves or pulses that overlap and can result in a combination of both constructive and destructive interference
  • For example, the peak of one wave superposes with the peak of another wave with a smaller displacement
  • The resultant peak will have a displacement that is in the middle of the displacement of both waves

• Superposition can also be demonstrated with two pulses
  • When the pulses meet, the resultant displacement is the algebraic sum of the displacement of the individual pulses
  • After the pulses have interacted, they then carry on as normal

• Interference occurs whenever two or more waves combine to produce a resultant wave with a new resultant displacement
  • The waves combine according to the principle of superposition

• When constructive interference occurs:

Two coherent waves combine by adding their displacements to create a resultant wave with a larger amplitude

• When destructive interference occurs:

Two coherent waves combine by cancel out their displacements to create a resultant wave with a smaller amplitude

• The type of interference occurring at a given point (i.e. constructive or destructive) depends on the path difference of the overlapping waves

Path Difference

• Path difference is defined as:

The difference in distance travelled by two waves from their sources to the point where they meet

• Path difference is generally expressed in multiples of wavelength

At point P₂ the waves have a path difference of a whole number of wavelengths resulting in constructive interference. At point P₁ the waves have a path difference of an odd number of half wavelengths resulting in destructive interference 

• In the diagram above, the number of wavelengths between:
  • S₁ ➜ P₁ = 6λ
  • S₂ ➜ P₁ = 6.5λ
  • S₁ ➜ P₂ = 7λ
  • S₂ ➜ P₂ = 6λ

• The path difference is:
  • (6.5λ – 6λ) =  at point P₁
  • (7λ – 6λ) = λ at point P₂

• Hence:
  • Destructive interference occurs at point P₁
  • Constructive interference occurs at point P₂

Conditions for Constructive and Destructive Interference

• In general, for waves emitted by two coherent sources very close together:
• The condition for constructive interference is:

path difference = 

• The condition for destructive interference is:

path difference =

• Where:
  • λ = wavelength of the waves in metres (m)
  • n = 0, 1, 2, 3... (any other integer)

• The same conditions apply to waves emitted by a single coherent source and diffracted by two narrow slits very close together

Path Difference and Wavefronts

At point P the waves have a path difference of a whole number of wavelengths resulting in constructive interference

• Another way to represent waves spreading out from two sources is shown in the diagram above
• At point P, the number of crests from:
  • Source S₁ = 4λ
  • Source S₂ = 6λ

• The path difference at P is (6λ – 4λ) = 2λ
• This is a whole number of wavelengths (n = 2), hence constructive interference occurs at point P