Physics PHY9702
16. SUPERPOSITION
Phase difference between the two waves at point P
π
Source 2
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Physics PHY9702
16.1 Stationary waves (Standing waves) A stationary or standing wave is one in which the amplitude varies from place to place along the wave. Figure 1 is a diagram of a stationary wave. Note that there are places where the amplitude is zero and, halfway between, places where the amplitude is a maximum; these are known as nodes (labelled N) and antinodes (labelled A) respectively. (See Figure 1)
The distance between successive nodes, and successive antinodes, is half a wavelength. (λ/2) The amplitude of the points on a stationary wave varies along the wave. In Figure 1 the amplitude at point 1 is a1, that at point 2 is a2 and that at point 3 is a3. The displacement (y) at these points varies with time. Any stationary wave can be formed by the addition of two travelling waves moving in opposite directions. A wave moving in one direction reflects at a barrier and interferes with the incoming wave.
Mathematical treatment of the formation of a standing wave from two travelling waves Consider two travelling waves 1 and 2. Let the displacements at time t and position x be y1 and y2. y1 = a sin (ωt - kx) (say right- left) y2 = a sin (ωt + kx) (say left- right) Therefore:
Note that this expression is composed of two terms: (a) sin (ωt) - this shows a varying amplitude with time at a particular place. (b) cos (kx) - this shows a varying amplitude with position at a particular time.
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When x = 0, λ/2 ... A is a maximum and we have an antinode; When x = λ/4, 3λ/4, 5λ/4 ... A is a minimum and we have a node. Notice that the maximum value of A is 2a.
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Stationary waves in strings
L Fundamental (1st harmonic)
λ =L 2
L 1st overtone (2nd harmonic)
λ=L
L 2nd overtone (3rd harmonic)
3λ =L 2
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Stationary wave in pipes (tubes) The Physics of sound in tubes This part of the Physics of sound is the basis of all wind instruments, from the piccolo to the organ. Basically the ideas are very simple but they can become complex for a specific musical instrument. For that reason we will confine ourselves to a general treatment of the production of a note from a uniform tube. The stationary waves set up by the vibrations of the air molecules within the tube are due to the sum of two travelling waves moving down the tube in opposite directions. One of these is the initial wave and the other its reflection from the end of the tube. All air-filled tubes have a resonant frequency and if the air inside them can be made to oscillate they will give out a note at this frequency. This is known as the fundamental frequency or first harmonic. Higher harmonics or overtones may also be obtained and it is the presence of these harmonics that gives each instrument its individual quality. A note played on a flute will be quite unlike one of exactly the same pitch played on a bassoon! A harmonic is a note whose frequency is an integral multiple of the particular tube's or string's fundamental frequency. Tubes in musical instruments are of two types: (a) open at both ends, or (b) open at one end and closed at the other. The vibration of the air columns of these types of tube in their fundamental mode are shown in Figure 1. Notice that the tubes have areas of no vibration or nodes at their closed ends and areas of maximum vibration or antinodes at their open ends. An antinode also occurs at the centre of a tube closed at both ends in this mode. Nodes are areas where the velocity of the molecules is effectively zero but where there is a maximum variation in pressure, while the reverse is true for antinodes. Some of the higher harmonics for the different tubes are shown in Figure 2. Notice that a closed tube gives odd-numbered harmonics only, while the open tube will give both odd and even-numbered.
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Notice that although the sound waves in the tubes are longitudinal it is conventional to represent them as transverse vibrations for simplicity. However two examples of what are really going on is given for completeness.
If the velocity of sound is denoted by v and the length of a tube by L, then for a tube closed at one end the fundamental frequency is given by:
For a tube open at both ends the fundamental frequency is given by
End corrections The vibrations within the tube will be transmitted to the air just outside the tube, and the air will then also vibrate. In accurate work we must also allow for this effect, by making an end correction (Figure 3). This means that we consider that the tube is effectively longer than its measured length by an amount d, that is: The true length = L ± d. The equation for a closed tube then becomes:
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Measurement of the velocity of sound The velocity of sound in air may be found quite simply by using the resonance of a column of air in a tube. An open- ended tube is placed in a glass cylinder containing water, as shown in Figure 4, so that the water closes the bottom end of the tube. A tuning fork of known frequency is sounded over the upper end, the air in the tube vibrates and a note is heard. The length of the air column is adjusted by raising the tube out of the water until a point is found where resonance occurs and a loud note is produced. At this point the frequency of the tuning fork is equal to the resonant frequency of the tube. In its fundamental mode the wavelength A is four times the length of the air column (L), that is: λ= 4L Since velocity = frequency x wavelength the velocity of sound may be found. For accurate determinations the following precautions should be taken: (a) the temperature of the air should be taken, since the velocity of sound is temperature-dependent, and (b) the end correction should be allowed for. This may be done by finding the resonance for the second harmonic with the same tuning fork.
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Vibrating strings
If a string stretched between two points is plucked it vibrates, and a wave travels along the string. Since the vibrations are from side to side the wave is transverse. The velocity of the wave along the string can be found as follows. Velocity of waves along a stretched string Assume that the velocity of the wave v depends upon (a) the tension in the string (T), (b) the mass of the string (M) and (c) the length of the string (L) (see Figure 1). Therefore: v = kTxMyLz Solving this gives x = ½ , z = ½ , y = - ½ . The constant k can be shown to be equal to 1 in this case and we write m as the mass per unit length where m = M/L. The formula therefore becomes:
Since velocity = frequency x wavelength
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The Physics of vibrating strings A string is fixed between two points. If the centre of the string is plucked vibrations move out in opposite directions along the string. This causes a transverse wave to travel along the string. The pulses travel outwards along the string and when they reaches each end of the string they are reflected (see Figure 2).
The two travelling waves then interfere with each other to produce a standing wave in the string. In the fundamental mode of vibration there are points of no vibration or nodes at each end of the string and a point of maximum vibration or antinode at the centre. Notice that there is a phase change when the pulse reflects at each end of the string. The first three harmonics for a vibrating string are shown in the following diagrams. (a) As has already been shown; for a string of length L and mass per unit length m under a tension T the fundamental frequency is given by:
(b) First overtone or second harmonic:
(c) Second overtone or third harmonic:
A string can be made to vibrate in a selected harmonic by plucking it at one point (the antinode) to give a large initial amplitude and touching it at another (the node) to prevent vibration at that point.
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16.2 Diffraction When a wave hits an obstacle it does not simply go straight past, it bends round the obstacle. The same type of effect occurs at a hole - the waves spread out the other side of the hole. This phenomenon is known as diffraction and examples of the diffraction of plane waves are shown in Figure 1.
The effects of diffraction are much more noticeable if the size of the obstacle is small (a few wavelengths across), while a given size of obstacle will diffract a wave of long wavelength more than a shorter one.
Diffraction can be easily demonstrated with sound waves or microwaves. It is quite easy to hear a sound even if there is an obstacle in the direct line between the source and your ears. By using the 2.8 cm microwave apparatus owned by many schools very good diffraction effects may be observed with obstacles a few centimetres across. One of the most powerful pieces of evidence for light being some form of wave motion is that it also shows diffraction. The problem with light, and that which led Newton to reject the wave theory is that the wavelength is very small and therefore diffraction effects are hard to observe. You can observe the diffraction of light, however, if you know just where to look.
The coloured rings round a street light in frosty weather, the coloured bands viewed by reflection from a record and the spreading of light round your eyelashes are all diffraction effects. Looking through the material of a stretched pair of tights at a small torch bulb will also show very good diffraction. A laser will also show good diffraction effects over large distances because of the coherence of laser light.
Diffraction is essentially the effect of removing some of the information from a wave front; the new wave front will be altered by the obstacle or aperture. Huygens' theory explained this satisfactorily.
Grimaldi first recorded the diffraction of light in 1665 but the real credit for its scientific study must go to Fresnel, Poisson and Arago, working in the late eighteenth and early nineteenth centuries.
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Fresnel and Fraunhofer diffraction We can define two distinct types of diffraction: (a) Fresnel diffraction is produced when light from a point source meets an obstacle, the waves are spherical and the pattern observed is a fringed image of the object. (b) Fraunhofer diffraction occurs with plane wave-fronts with the object effectively at infinity. The pattern is in a particular direction and is a fringed image of the source.
Fresnel diffraction Fresnel diffraction can be observed with the minimum of apparatus but the mathematics are complex. We will therefore only treat it experimentally here. If a razor blade is placed between the observer and a point source of monochromatic light, dark and bright diffraction fringes can be seen in the edges of the shadow. The same effects can be produced with a pinhead, when a spot of light will be seen in the centre of the shadow. Fresnel was unhappy about Newton's explanation of diffraction in terms of the attraction of the light particles by the particles of the solid, because diffraction was found to be independent of the density of the obstacle: a spider's web, for example, gave the same diffraction pattern as a platinum wire of the same thickness. The prediction and subsequent discovery of a bright spot within the centre of the shadow of a small steel ball was final proof that light was indeed a wave motion.
the intensity of light is plotted against distance for points close to the shadow edge results like those shown in Figure 2 will be obtained.
If
Fresnel diffraction with a double slit will produce two single slit patterns superimposed on one another. This is exactly what happens in the Young's slit experiment: the diffraction effects are observed as well as those due to the interference of the two sets of waves.
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Fraunhofer diffraction - Single slit
The Fraunhofer diffraction due to a single slit is very easy to observe. An adjustable slit is placed on the table of a spectroscope and a monochromatic light source is viewed through it using the spectroscope telescope (see Figure 1(a)). An image of the slit is seen as shown in Figure 1(b). As the slit is narrowed a broad diffraction pattern spreads out either side of the slit, only disappearing when the width of the slit is equal to or less than one wavelength of the light used. The diffraction at a single slit of width a is shown in Figure 2. Diffraction occurs in all directions to the right of the slit but we will just concentrate on one direction towards a point P in a direction θ to the original direction of the waves. Plane waves arrive at P due to diffraction at the slit AB. Waves coming from the two sides of the slit have a path difference BN and therefore interference results. But BN = a sin(θ), and if this is equal to the wavelength of the light (λ) the light from the top of the slit and the bottom of the slit a will cancel out.and a minimum is observed at P. This is because if the path difference between the two extremes of the slit is exactly one wavelength there will be points in the upper and lower halves of the slit that will be half a wavelength out of phase. Therefore the general condition for a minimum for a single slit is:
The path difference between light from the top and bottom of the slit is written mλ where m is the number of wavelengths 'fitting into' BN. m is also known as the 'order' of the diffraction image. If the intensity distribution for a single slit is plotted against distance from the slit, a graph similar to that shown that shown in Figure 3 will be obtained. The effect on the pattern of a change of wavelength is shown in Figure 4.
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Wavelength effects These two diagrams show the effect of a change of wavelength on the single slit diffraction pattern. The pattern for red light is broader than that for blue because of the longer wavelength of red light.
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Fraunhofer diffraction - double slit For the double slit we simply have light from two adjacent slits meeting at the eyepiece. In this case the formula for a maximum (a place where the light waves 'add up') is:
where d is the distance between the centres of the two slits (See Figure 1). The intensity of the interference pattern produced by two sources is simply varied by the diffraction effects. We will have cos2 fringes modulated by the diffraction pattern for a single slit. The intensity distribution is shown in Figure 2.
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16.3 Interference Interference: Coherence: In physics, coherence is a property of waves, that enables stationary (i.e.
temporally and spatially constant) interference. More generally, coherence describes all correlation properties between physical quantities of a wave. When two groups of waves (called wave trains) meet and overlap they interfere with each other. The resulting amplitude will depend on the amplitudes of both the waves at that point. If the crest of one wave meets the crest of the other the waves are said to be in phase and the resulting intensity will be large. This is known as constructive interference. If the crest of one wave meets the trough of the other (and the waves are of equal amplitude) they are said to be out of phase by π then the resulting intensity will be zero. This is known as destructive interference. This phase difference may be produced by allowing the two sets of waves to travel different distances - this difference in distance of travel is called the path difference There will be many intermediate conditions between these two extremes that will give a small variation in intensity but we will confine ourselves to total constructive or total destructive interference for the moment. The diagrams in Figure 1 below show two waves of equal amplitudes with different phase and path differences between them. The first pair have a phase difference of π or 180o and a path difference of an odd number of half-wavelengths. The second pair have a phase difference of zero and a path difference of a whole number of wavelengths, including zero.
Figure 1(a) shows destructive interference and Figure 1(b) constructive interference. To obtain a static interference pattern at a point (that is, one that is constant with time) we must have (a) two sources of the same wavelength, and (b) two sources which have a constant phase difference between them. Sources with synchronised phase changes between them are called coherent sources and those with random phase changes are called incoherent sources. This condition is met by two speakers connected to a signal generator because the sound waves that they emit are continuous – there are no breaks in the waves.
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Physics PHY9702 However two separate light sources cannot be used as sources for a static interference pattern because although they may be monochromatic the light from them is emitted in a random series of pulses of around 10-8 s duration. The phase difference that may exist between one pair of pulses emitted from the source may well be quite different from that between the next pair of pulses (Figure 2).
Therefore although an interference pattern still occurs, it changes so rapidly that you get the impression of uniform illumination. Another problem is that the atoms emitting the light may collide with each other so producing phase changes within one individual photon. We must therefore use one light source and split the waves from it into two in some way. There are two ways of doing this: (a) division of amplitude, where the amplitude at all points along the wavefront is divided between the two secondary waves, and (b) division of wavefront, where the original wave-front is divided in two, half of it forming each of the secondary waves. However, the length of each pulse limits the path difference that we may obtain between even these two waves from the same source. Since the pulses are only about 10-8 s long the maximum path difference is 3 m, although in practice good results are only obtained with shorter path differences than this.
Interference between two waves The diagrams in Figure 3-7 show two sources S1 and S2 emitting waves - they could be light, sound or microwaves. The plan view of the waves in Figure 3 shows waves coming from two slits and interfering with each other. The lines along which the path differences will give maxima or minima. This type of arrangement is like that produced in a ripple tank or in the double slits experiment with light (see later). It should be realised that between the maxima and minima the intensity will change gradually from one extreme to the other.
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Figure 4(a) shows light interfering as it passes through two slits. In Figure 4(a) the appearance of the interference pattern on a screen placed in the path of the beam is shown. You can see the maxima and minima and the way in which the intensity changes from one to the other. Changing the wavelength of the light (its wavelength), the separation of the slits or the distance of the slits from the screen will all give changes in the separation of the maxima in the interference pattern.
Figure 5 shows the interference effects of two speakers. The sound waves spread out all round the speakers and a static interference pattern is formed. (Not all the maxima and minima are labelled). You can hear this by setting up two speakers in the lab connected to one signal generator and then simply walking round the room. You will hear the sound go from loud to soft as you pass from maximum to minimum. (A frequency of around 400 Hz is suitable).
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In Figures 6 and 7 you can see that at the different points on the screen the waves from S1 have travelled a different distance from those from S2. In Figure 6 the path difference is zero, in Figure 7 it is half a wavelength.
Path length in a material When light passes through a material of refractive index n it is slowed down, its velocity in the material being 1/n times that in a vacuum. For example, the velocity of light in glass is about 2.0x108 ms-1 compared with about 3.00 x 108 ms-1 in a vacuum. The time light takes to pass through a given length of the material is therefore n times that which it takes to pass through the same length of air. The path length in a material of length L and refractive index n is therefore nL (Figure 8). If one part of a light beam travels a distance L in air and the other a distance L in the material then a path difference will exist between them of L(n 1) and if the two beams are made to overlap an interference pattern will result.
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16.4 Two-source interference patterns Patterns which are produced when two circular periodic waves interfere are often termed a two source interference pattern.
Young's double slit experiment This experiment to use the effects of interference to measure the wavelength of light was devised by Thomas Young in 1801, although the original idea was due to Grimaldi. The method produces non-localised interference fringes by division of wavefront, and a sketch of the experimental arrangement is shown in Figure 1.
L igh t
from a monochromatic line source passes through a lens and is focused on to a single slit S. It then falls on a double slit (S1 and S2) and this produces two wave trains that interfere with each other in the region on the right of the diagram. The interference pattern at any distance from the double slit may be observed with a micrometer eyepiece or by placing a screen in the path of the waves. The separation across double slit should be less than 1 mm, the width of each slit about 0.3 mm, and the distance between the double slit and the screen between 50 cm and 1 m. The single slit, the source and the double slit must be parallel to produce the optimum interference pattern. Alternatively a laser may be used and the fringes viewed on a screen some metres away without the need for a micrometer eyepiece or a single slit.The formula relating the dimensions of the apparatus and the wavelength of light may be proved as follows.
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Consider the effects at a point P a distance xm from the axis of the apparatus. The path difference at P is S2P - S1P. For a bright fringe (constructive interference) the path difference must be a whole number of wavelengths and for a dark fringe it must be an odd number of halfwavelengths (Figure 2). Consider the triangles S1PR and S2PT.
S1P2= (xm – d/2)2 + D2 S2P2 = (xm2 + d/2)2 + D2 Therefore: S2P2 – S1P2 = 2xmd so (S2P - S1P)(S2P + S1P) = 2xmd But S2P + S1P = 2D within the limits of experimental accuracy for D would be at least 50 cm while d would be less than 1 mm making the triangle S1S2P very thin. Therefore: S2P – S1P = xmd/D
Where m = 0,1,2,3 etc. and so the m th bright fringe for m = 3 is 3lD/d from the centre of the pattern. The distance between adjacent bright fringes is called the fringe width (x) and this can be used in the equation as:
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Note that the fringe width is directly proportional to the wavelength, and so light with a longer wavelength will give wider fringes. Although the diagram shows distinct light and dark fringes, the intensity actually varies as the cos2 of angle from the centre. If white light is used a white centre fringe is observed, but all the other fringes have coloured edges, the blue edge being nearer the centre. Eventually the fringes overlap and a uniform white light is produced.
The separation of the two slits should be of the same order of magnitude as the wavelength of the radiation used.
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16.5 Diffraction gratings If the number of slits in an obstacle is now increased we will see that the sharpness of the pattern is improved, the maxima getting narrower. Obstacles with a large number of slits (more than, say, 20 to the millimetre) are called diffraction gratings. These were first developed by Fraunhofer in the late eighteenth century and they consisted of fine silver wire wound on two parallel screws giving about 30 obstacles to the millimetre. Since then many improvements have been made, in 1882 Rowland used a diamond to rule fine lines on glass, the ridges acting as the slits and the rulings as the obstacles (Figure 1). Using this method it is possible to obtain diffraction gratings with as many as 3000 lines per millimetre although 'coarse' gratings with about 500 lines per millimetre are better for general use. In many schools two types are in common use, one with 300 lines per mm and the other with 80 lines per mm. Reflection gratings are also used, where the diffracted image is viewed after reflection from a ruled surface. A very good example of a reflection diffraction grating is a CD. A DVD with finer rulings gives a much broader diffraction pattern.
The wave theory and the diffraction grating
Figure 2 shows the Huygens construction for a grating. You can see how the circular diffracted waves from each slit add together in certain directions to give a diffracted wave which has a plane wave front just like the waves hitting the grating from the left. This plane wave is formed by drawing the line that meets all the small circular waves and is called an envelope of all these small secondary waves.
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The diffraction grating formula Consider a parallel beam of light incident normally on a diffraction grating with a grating spacing e (the grating spacing is the inverse of the number of lines per unit length). Consider light that is diffracted at an angle θ to the normal and coming from corresponding points on adjacent slits (Figure 3). For a maximum the path difference = AC = mλ But AC = e sinθ. Therefore for a maximum:
The number m is known as the order of the spectrum, that is, a first-order spectrum is formed for m = 1, and so on. If light of a single wavelength, such as that from a laser, is used, then a series of sharp lines occur, one line to each order of the spectrum. With a white light source a series of spectra is formed with the light of the shortest wavelength having the smallest angle of diffraction. In deriving the formula above, we assumed that the incident beam is at right angles to the face of the grating. Allowance must be made if this is not the case. The simplest way is to measure the position of the first order spectrum on either side of the centre, record the angle between these positions and then halve it, as shown in Figure 4. The number of orders of spectra visible with a given grating depends on the grating spacing, more spectra being visible with coarser gratings. The ruled face of the grating should always point away from the incident light to prevent errors due to changes of direction because of refraction in the glass. The diagram shows a central white fringe with three spectra on either side giving a total of seven images. (See example problem )
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The intensity distribution for a large number of slits is shown in Figure 5. Notice that the maxima become much sharper; the greater the number of slits per metre, the better defined are the maxima.
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