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WAVES

SUPERPOSITION PRINCIPLE

Huygens Principle

Diffraction and Interference

 

 

 

 

 

Particles and waves carry energy from one place to another. Both particles and waves can be reflected and refracted.

 

So, what characteristics distinguish waves from particles?

 

Two rocks cant be in the same place at the same time. However, multiple waves can occupy the same space at the same time. When more than one wave occupies the same space, at each instant a resultant waveform is produced which is simply the algebraic sum of the individual wavefunctions. This concept is known as the Superposition Principle. For example, two pulses travelling in opposite directions will pass through each other unaffected, while passing, through each other, the resultant displacement is simply the sum of the individual displacements.

 

The superposition principle can be used to describe the interference and diffraction behaviour of waves. In the classical view of the universe, particles do not obey the superposition principle, hence, particles do not interfere with each other and do not undergo diffraction. But our universe is much more interesting at the atomic level. Using the concepts of quantum mechanics, waves have particle like properties and particles have wave like properties. This is known as the Wave-Particle Duality Principle. Experiments show that particles such as electrons can be assigned a wavelength and can produce diffraction patterns. These ideas will be explored more deeply in later Modules.

 

Applying the superposition principle when the crest of one wave overlaps the crest of another wave, their individual effects add together to produce a wave of greater amplitude. This is called constructive interference. When the crest of one wave overlaps the trough of another, the amplitude of the resultant wave is reduced. This is called destructive interference.

 

 

 

Thinking exercise Predict Observe Explain POE

 

Carefully view the set of animations of the overlapping two [1D] sinusoidal waves travelling in opposite directions and interfering with each. The equation describing the two wave functions are

 

 

 

 

The speed of propagation is fixed by the properties of the medium, therefore,

 

 

 

Therefore, the wavefunctions #1, #2 and resultant wavefunction can be expressed as

 

 

 

Predict how the two waves interfere with each other, Observe the animations and Explain any discrepancies.

 

For each animation, estimate for wave #1, wave #2 and the resultant wave (wave #3): amplitude, wavelength, period, frequency and speed of propagation.

 

 

 

 

 

 

 

Animation 1.

 

 

 

 

Animation 2.

 

 

 

 

Animation 3.

 

 

 

POE summary

The two pulses travelling in opposite directions will pass through each other unaffected, while passing, through each other, the resultant displacement is simply the sum of the individual displacements.

 

Animation 1.

wave #1

wave #2

wave #

 

You get a surprising result when the two waves have identical speeds, amplitudes, wavelengths and periods. The waves are steadily in and out of phase with each other. The waveform is stationary the speed of propagation is zero. Such a wave is called a standing wave (stationary wave). There are nodes where there is complete cancellation with zero energy and antinodes where there is a maximum reinforcement of the waves and maximum energy. For a standing wave, the positions of the nodes and antinodes is fixed, that is, their positions are independent of time.

 

Standing waves are setup in the strings of musical instruments when plucked, bowed or struck. The initial disturbance travels along the string and reflected at the ends. Standing waves are formed at certain frequencies of vibration by the reflections of the waves as they move backward and forward along the string and interfere with each other. Standing waves are formed in wind musical instruments such organ pipes, trumpets, and clarinet by reflections of the sound waves in the pipe of the instruments.

 

Animation 2.

wave #1

wave #2

wave #3

 

The resultant wave is a sinusoidal wave moving to the right at 2.0 m.s-1.

 

Animation 3.

wave #1

wave #2

wave #3

 

The resultant wave is a complex wave moving to the right at 2.0 m.s-1.

 

 

 

Huygens Principle

 

When you make a sound in the kitchen, the sound spreads in all directions away from you. It's easy to think about the sound waves as if they just move in a straight line but this is not true another person in the dining room can hear the sound from the kitchen the wave heading toward the door into the dining room goes through that doorway, and again spreads in all directions enabling you to hear the sound.

 

This behaviour of waves was studies in 1678 by the Dutch physicist, Christian Huygens.

 

Huygens Principle - every point of a wavefront may be considered the source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the waves.

Fig. 1. Huygens principle can be used to determine the path of a wavefront (plane wave, circular wave, reflected wave, refracted wave, etc) by drawing a set of arcs of circles from the old to the new wavefront.

 

 

 

Diffraction

 

Any bending of light by means other than reflection or refraction is called diffraction. Diffraction occurs when a wave passes through an opening, around an edge or around an obstacle. Figure 2 shows sketches for the diffraction due to an opening and an edge.

 

Fig. 2. A wave bends and spreads as it passes an edge.

Why can we hear around corners, but cant see around corners?

 

Fig.3. When waves bend around an edge, they can also interfere. A photograph of a razor blade illuminated by a monochromatic (single wavelength) source. Notice the interference fringes around the outline of the blade.

 

 

The amount of diffraction depends on the wavelength of the wave compared with the size of the obstacle that casts a shadow as shown in figure 4. Wavelengths much greater the dimensions of the obstacle diffract more - they fill in the shadow regions. On the other hand, for waves with wavelengths much smaller than the obstacle, there is little bending and there are definite shadow regions.

 

Fig. 4. The amount of diffraction depends on the wavelength of the wave compared with the size of the obstacle that casts a shadow.

AM radio signals are relative long wavelengths (~200 to ~600 m) and these waves readily bend around buildings and other obstructions. A long-wavelength radio wave doesnt see a relatively small building in its path, but a short wavelength radio wave does. The FM band radio waves have short wavelength (2.8 to 3.4 m) and dont bend around buildings very well. Driving through the city, you will notice a AM radio station has a much better reception than an FM radio station.

 

When a wave passes through an aperture, the wave spread because each point on the wavefront can be consider a set of point sources, each radiating in all directions beyond the aperture this is diffraction. This causes serious problems for telescopes and microscopes. So, when you use a telescope to image two stars close together, the light entering the microscope is diffracted a point object becomes a blurred circle and this makes it difficult to distinguish two stars as separate identities if they are too close together.

 

The image of two stars?

 

 

 

 

 

 

In viewing an image with a microscope, if the size of the object is about the same as the wavelength of light, diffraction blurs the image of the object. If the object is smaller than the wavelength, no structure can be seen the image is lost because of diffraction. To obtain clear images of very small objects, electron microscopes are used. Highly energetic electrons (electron do have wave-like properties) have much smaller wavelengths than visible light and so very small objects can be imaged. Magnetic fields and electric fields are used as lenses to focus electron beams to get the highly magnify images of small objects.

 

 

 

 

 

Modelling [2D] Waves: Interference

 

We can model [2D] waves (e.g. surface waves on water) to visual their behaviour through a set of animations. A number of mechanical vibrators produce disturbances at a fixed frequency. The waves then spread in all directions from the sources. The speed of propagation of the waves is constant and is determined by the properties of the medium. For the animations, the frequency of the sources is set at 100 Hz and the propagation speed is 25 m.s-1.

 

 

 

 

Animation 4. A point source produces waves spreading in all directions with circular wavefronts. The source frequency is 100 Hz (T = 0.0100 s). The speed of propagation is 25.0 m.s-1 and the wavelength is 0.25 m. The green regions are the crests and the blue are the troughs. In the intensity plot, red is high intensity and yellow low. Notice that the intensity (and amplitude) of the wave decreases away from the source. In [3D] the intensity falls as 1/r2 (amplitude falls as 1/r) inverse square law. The source is located at the origin (0, 0). The amplitude of the wave from the point source goes to infinity as the distance between the observation point and the source goes to zero. So, in the plots, the wave amplitude can only be calculated for points well away from the source point.

 

 

 

 

We will now consider two sources. In some places crests overlap crests (constructive interference) and in other places crests and troughs overlap (destructive interference).

 

 

Animation #5 shows the interference pattern from two sources.

 

 

Animation 5A. The spacing of the sources is (). The bottom plot shows the intensity pattern for the two sources. The yellow shows the regions for the nodes nodal lines. The intensity pattern clearly shows the formation of distinctive antinodal and nodal lines set up by the two sources. Location of sources: (-0.5 m, 0) and (-0.5 m, 0).

 

 

Animation 5B. The spacing of the sources is (). The two sources are closer together than in animation 5A, but the spacing of the nodal lines is much greater. This type of relationship is true for diffraction and interference the smaller the dimensions (e.g. source separations, aperture sizes, etc) the greater the spreading of the nodal lines. The intensity graph uses a log scaling for the colour to show the nodal lines given in yellow.

Location of sources: (-0.1 m, 0) and (-0.1 m, 0).

 

        Sound (two speakers): if you walked along a nodal line, the sound would be very quiet but very loud along an antinodal line.

        Light (two red lasers): nodal regions would be dark and antinodal regions bright.

 

 

 

 

 

 

Diffraction in the real world

 

A key characteristic of a crystal is that it has a regular repeating structure of molecules arranged in planes. When such a crystal is exposed to electromagnetic radiation a diffraction pattern can be generated provided the wavelength of the radiation is comparable to the spacing d between the parallel planes within the crystal. Typically, , but the wavelength of visible light is from approximately 400 to 750 nm. Therefore, visible light will not produce a useful diffraction pattern from a crystal. However, X-rays have wavelengths ~ 0.1 nm and vivid diffraction patterns can be produced. A great deal of information about the arrangement and spacing between molecules can be extracted the X-ray diffraction patterns. J.D. Watson and F.H.C. Crick used X ray diffraction analysis in 1953 to help them deduce the double-helix structure of DNA.

 

Fig. 6. X ray diffraction.

 

 

Destructive / constructive interference plays a key role in the operation of a CD. A series of bumps are encoded on a smooth reflecting surface. A laser beam is reflected off the reflective surface to a detector. As a bump moves through the laser beam, the detector either receives a weak off signal or a strong on signal depending upon the whether the reflections are out of phase (destructive interference) or in phase (constructive interference) as shown in figure 13.

 

Fig. 7. Reading information on a CD.

 

 

 

 

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If you have any feedback, comments, suggestions or corrections please email:

Ian Cooper School of Physics University of Sydney

ian.cooper@sydney.edu.au