SuperpositionSuperposition

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

If two progressive waves are in phase then the maximum positive displacements from each wave line up, creating a resultant displacement with an increased amplitude (constructive interference). If two progressive waves are in antiphase, then the maximum positive displacement from one wave lines u with the maximum negative displacement of another, and the resultant displacement is smaller than for each individual wave. This is called destructive interference.

Coherent: refers to waves emitted from two sources having a constant phase different. Have the same frequency.
Interference: waves superposing with each other in a constructive or destructive way. Constructive interference produces maxima; destructive interference leads to minima.
Path difference: difference in distance travelled from the source. If the path difference is λ, where n is an integer, the two waves will arrive at that point in phase so constructive interference will occur. If the path difference is ( + ½)λ, the two waves will arrive at that point in antiphase and so destructive interference will occur.

Phase difference: the difference between the displacements of particles along a wave, or the difference between the displacements of particles on different waves, where 360° is equivalent to one complete cycle. A phase difference of × 360° leads to constructive interference; a phase difference of ( + ½ ) 360° leads to destructive interference.

	Path difference	Phase difference
2^nd order maxima	2λ	4π = 720°
2^nd order minima	1.5λ	3π = 540°
1^st order maxima	Λ	2π = 360°
1^st order minima	0.5λ	π = 180°
Central maxima	0	0 = 0°
1^st order minima	0.5λ	π = 180°
1^st order maxima	Λ	2π = 360°
2^nd order minima	1.5λ	3π = 540°
2^nd order maxima	2λ	4π = 720°

Techniques and procedures for demonstrating the principle of superposition:

Sound: two loudspeakers connected to the same signal generator produce coherent sound waves. This creates an interference pattern comprising of a series of maxima (louder) and minima (softer). You can detect these with your ear or a microphone.

Microwaves: use a single microwave source and a pair of slits (double slit). Detect the interference pattern with a microwave receiver connected to a voltmeter or an oscilloscope.

Light: Young’s double slit experiment and diffraction grating.

Young’s double slit experiment: use two coherent waves with a monochromatic source of light (achieved using a colour filter that only allows a specific frequency of light to pass) and a narrow slit to diffract the light. Light diffracting from the single slit arrives at the double slit in phase, then diffracts again at the double slit. Each slit acts as a source of coherent waves which interfere, producing an interference pattern seen on the screen as alternating bright and dark regions called fringes. This demonstrates the wave nature of light.

The separation between slits ₁and ₂ is . . A bright fringe (central maxima) is seen at , and the next adjacent bright fringe is seen at . The separation between the fringes is . The path difference ₁ must be equal to one whole wavelength. The two rays of light are almost parallel to each other so ₁ and ₂ are almost the same and very small. Hence,

$sin(Theta _{1})approx sin(Theta _{2})approx tan(theta)_{2}$

$xsin(Theta _{1})=frac{lambda }{a} and sin(Theta _{2})=_{D-ax}$

(An alternative expression of this is: sin = λ).

This equation can be used to determine the wavelength of light using a double slit or diffraction grating. Note that the fringes get further apart if you: decrease a, decrease the wavelength of the light used, or increase D.