When two waves (of the same type) meet they sort of pass through each other/overlap. This is known as superposition and a single wave is produced whose instantaneous displacement is the sum of the two former waves (the principle of superposition reads: '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'). It is important to realise that, since displacement is a vector quantity, this resultant displacement can be bigger or smaller than the displacements of the previous waves. This effect is known as interference. If two waves are in phase the maximum positive displacements will line up causing constructive interference. This results in an increase in amplitude since intensity ∝ (amplitude)². If the two waves are in antiphase the maximum positive displacement of one will line up with the maximum negative displacement of another - this results in destructive interference. If the amplitudes of both waves are the same, the resultant will have zero amplitude:
Interference patterns can be seen when, for example, raindrops fall on a pond. As the waves ripple outward they overlap with waves caused by other drops. At this point they superpose and can interfere constructively (if in phase) or destructively (if out of phase). If is important to realise that this does not produce a stable interference pattern but rather one that changes all the time, for a stable pattern the waves must be coherent. Coherence refers to waves emitted from two sources that have a constant phase difference. For example, two filament lamps cannot produce stable interference patterns as they emit light of a range of different frequencies and changing phase difference between different waves (in other words, they do not produce coherent light).
Interference patterns contain a series of fringes known as maxima (louder/brighter etc, where constructive interference occurs) and minima (quieter/dimmer etc, where destructive interference occurs). Maxima and minima are the result of two waves that have travelled different distances from their sources. This difference is known as the path difference. If the path difference to a point is a whole number (or 0) the two waves will arrive in phase (producing constructive interference) producing a point that has maximum amplitude. If the path difference to a point is an odd number of half wavelengths (n+0.5 where n is an integer) the two waves will arrive in antiphase (producing destructive interference) producing a point that has minimum amplitude.
At the central maxima the path difference is zero so the phase difference (the difference between displacements of particles along a wave) is zero. At the first order maxima the path difference is one wavelength, so the phase difference is 360° (the peaks from the first wave line up with the peaks from the second wave so constructive interference occurs). At the first order minima the path difference is half a wavelength so the phase difference is 180° (the peaks from the first wave line up with the troughs from the second waves which results in destructive interference).
Experiments:
Light
One way to observe the interference patterns of light is to observe the pattern of coloured light on thin oil films on water. Basically, light reflecting off the bottom surface of the oil interferes with the light reflected off the top surface. If the thickness of the oil results in a path difference that is a non-integer half number of wavelengths of light the two sets of light wavs are out of phase and destructive interference occurs and the waves cancel out. The colours result from the different wavelengths in white light and the differences in the thickness of the oil layer. The distance the light travels through the oil before reflecting off the surface differs. Different wavelengths of light are cancelled out by different thicknesses of oil. The wavelengths are not cancelled out from the colours we observe.
Credit: Kerboodle OCR Physics A textbook
Sound
- Connect two loudspeakers to the same signal generator (this means they will emit coherent sound waves)
- the sound waves will travel out from each loudspeaker and overlap forming an interference pattern
- this interference pattern comprises a series of maxima (louder areas) and minima (quieter areas)
- the positions of maxima and minima can be detected with a microphone (or your ears, but a microphone is more accurate)
Microwaves
- Introduce a pair of slits in front of a single microwave source
- the microwaves will diffract and overlap forming an interference pattern
- the interference pattern can be detected using a microwave receiver connected to a voltmeter or an oscilloscope
- if the receiver were to me moved in an arc around the double slit the maxima and minima created as part of the interference pattern can be detected
- the positions of each maxima and minima can be marked on a piece of paper situated below the apparatus
The Young double-slit experiment
As we know from above we need two coherent waves to form an interference pattern. Young used a monochromatic light source (by using a light filter). This means only light of a specific frequency can pass through a narrow slit that follows to diffract the light. Light diffracting from this slit then arrives at a double slit in phase. It 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 fringes (alternating bright and dark regions). This experiment demonstrates the wave nature of light - it can also determine the wavelengths of various different colours of visible light.
Okay so now for some maths. The separation between the double slits is denoted as 'a'. The interference pattern is observed on a screen at distance 'D'from the slits (D>>a). A bright fringe (maxima) is seen on the screen at position 'Y' and the next an position 'X', the distance between 'Y' and 'X' is x. The path difference S1P is one wavelength and the angles θ1 and θ2 are almost the same (they are very VERY small). We can use trig to show that:
sinθ1 ≈ sinθ2 ≈ tanθ2
where sinθ1 = wavelength/a and tanθ2 = x/D
This means that wavelength/a ≈ x/D. We can roughly express this as:
λ = ax/D
Provided D>>a.
Isaac Newton had a theory in which light was made up of tiny particles. Christiaan Huygens (a Dutch physicist) believed that light was made up of waves vibrating up and down perpendicular to the direction of the light travels. From this he formulated a way of visualising wave propagation (known as 'Huygens' Principle'). Huygens theory was the successful theory of light showing wave motion in three dimensions. He suggested that in a vacuum, or other uniform mediums, the light waves are spherical, and these wave surfaces advance or spread out as they travel at the speed of light. This theory explains why light shining through a pin hole or slit will spread out rather than going in a straight line (diffraction). Huygens theory better describes early experiments. Huygens' principle lets you predict where a given wavefront will be in the future, if you have the knowledge of where the given wavefront is in the present.
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