The principle of superposition

​​In a nutshell

Superposition occurs when waves overlap and combine to form a resultant wave at that point. When two waves continuously pass through each other, interference occurs. Interference can be constructive or destructive. Path difference is the difference in distance travelled by two waves from a source.

Equations

Variable definitions

Superposition

Superposition occurs when waves overlap and combine to form a resultant wave. This single wave only occurs at the point the waves meet and disappears when the waves have passed through each other. 

The principle of superposition states that when waves overlap, the resultant displacement is equal to the vector sum of the individual displacements.

Interference

Interference occurs when two progressive waves continuously overlap and combine to form a resultant wave. The displacement of this wave follows the principle of superposition for each point on the wave. It is equal to the sum of the individual displacements of the two waves.

A clear interference pattern can only occur if the two waves are coherent. This means that they have a constant phase difference, which can only happen if the two waves have the same frequency.

Note: ​Don't confuse constant phase difference with a phase difference of zero!

Constructive and destructive interference

There are two types of interference: constructive and destructive.

Constructive interference occurs when the waves are in phase. The maximum displacements in the same direction align (crests-to-crests and troughs-to-troughs) and they result in a displacement with a larger amplitude. This is known as a maxima.

As intensity is proportional to the square of the amplitude, a maxima will have an increased intensity. 

Destructive interference occurs when the waves are antiphase. The maximum displacements in opposite directions align (crests-to-troughs) and they result in a displacement with a smaller amplitude. This is known as a minima.

The intensity of a minima is decreased.

Path difference

The path difference between two waves is the difference in distance travelled from their source. Path difference is usually written in terms of wavelengths (e.g. $$1\lambda, \frac{3}{2}\lambda, \space$$ etc.)

The path difference between the two waves determine whether the wave will interfere constructively or destructively. 

If the path difference is an integer multiple of the wavelength ($$0, 1\lambda,2\lambda, ...$$) then the waves will interfere constructively and form a maxima. This is because it corresponds to a phase difference of an integer multiple of $$360 \degree$$ or $$2\pi\, rads$$ so the two waves are in phase.

​$$path \space difference = n\lambda \:\:\:\:\:\:\:\text{where }n\text{ is an integer}$$​​

If the path difference is an odd integer multiple of half wavelengths ($$\frac{1}{2}\lambda, \frac{3}{2}\lambda,...$$) then the waves will interfere destructively and form a minima. This is because it corresponds to a phase difference of an odd integer multiple of $$180\degree$$ or $$\pi \,rads$$ so the two waves are antiphase.

$$path \space difference = (n+\frac{1}{2})\lambda \:\:\:\:\:\:\:\text{where }n\text{ is an integer}$$

There will be constructive and destructive interference whenever the two waves meet. However, only specific path differences result in constructive or destructive interference which form maxima and minima.

Note: The value of $$n$$ corresponds to the $$n$$th order maxima or $$(n+1)$$th order minima. For example, when constructive interference occurs at a path difference of $$1\lambda$$, it is the first-order maxima. When destructive interference occurs at a path difference of $$\frac{1}{2}\lambda\:,(n=0),$$ it is the first order minima.

Interference of sound waves

The interference pattern of two speakers producing coherent sound waves placed at an equal distance from a vertical line can be worked out by looking at the path difference of the waves.

At points of constructive interference, the sound will be loud. At points of destructive interference, the sound will be quiet.

A microphone connected to an oscilloscope would be able to detect the intensity of the sound waves at each point. This would show how the interference changes along the vertical line.

Example

​The figure shows a pair of coherent speakers which produce a sound with a wavelength of $$4\,cm$$​. Point P lies $$0.96\,m$$​ from the first speaker and $$1.42\,m$$​ from the second speaker. With the help of a calculation, explain whether point P will be a minima or maxima.

First, write down the quantities given and check they are in the correct form:

The distance the sound wave travels from speaker A to point P, $$d_{AP}= 0.96\,m$$​

The distance the sound wave travels from speaker B to point P, $$d_{BP} = 1.42\,m$$

$$\lambda = 4\,cm = 0.04\,m$$​​

Next, write down the equations needed:

$$path \space diff = d_{BP} - d_{AP}$$​​

Then, substitute the values into the equations:

​$$path \space diff = 1.42 - 0.96 = 0.46\,m\newline path \space diff = \dfrac{23}{2}\lambda$$

The path difference is a half-integer multiple of the wavelength. This matches with the definition for the path difference for destructive interference:

$$path \space diff = (n+\frac{1}{2})\lambda \:\:\:\:\:\:\:\text{where }n\text{ is an integer}\newline \text{let}\space n = 11 \newline path\space diff = 0.46\,cm$$​​

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Therefore, destructive interference occurs at point P.