wavelength λ Amplitude of the wave A λ crest A trough observer speed of the wave v Period of the wave P λ = v P frequency f, f = 1/P v =λ f

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1 Waves Traveling waves: Traveling, periodic, sinusoidal (Shaped like a mathematical sine wave) water waves, sound waves, electromagnetic waves, and others, share certain features shown in the figure. The distance from crest to crest is the wavelength λ. They pass an oberver with speed v. The time between the passage of one crest and the next crest is the period P of the wave, so λ = vp. Traveling wave wavelength λ Amplitude of the wave A λ crest A trough observer Period of the wave P λ = v P v speed of the wave v frequency f, f = 1/P v =λ f The frequency f of a periodic wave is the reciprocal of the period, f = 1/P. So for example if P = 1/10 second, then f = 10 cycles/second. However, we use a different name for the unit to measure frequency. 1Hertz, abbreviated 1 Hz, is the same as 1 cycle/second. So, in the previous example, we say f = 10 Hz. Transverse Waves: One can imagine that the wave in the figure above is a transverse wave traveling on a rope. The wave travels to the right 1

2 but the motion of the rope is up and down, perpendicularly (transverse) to the direction of motion. Water ripples, shown in the figure below, and electromagnetic waves are also transverse. TRANSVERSE WAVES water wave: surface motion electromagnetic wave: Electric field amplitude wavefront Energy is carried along rays radius = r ray perpendicular to wavefront Water ripples travel on the surface of the water. Wave fronts are an important concept for waves. In this case the wave fronts are circles that follow the crests of the wave. Longitudinal Waves: Sound waves and waves on a spring, shown in the figure below, are longitudinal waves because the motion of the material particles is back and forth from left to right as the wave travels to the right. In sound through air it is the air molecules that move back and forth from left to right as the wave travels to the right. Sound waves also travel through solids and liquids. If the figure represents a sound wave, the compressed regions have higher pressure and density than the stretched regions. 2

3 In the spring regions of compression and stretching move back and forth from left to right as the wave travels to the right. v LONGITUDINAL WAVE a long spring vibrating as a logitudinal wave passes Standing waves or normal modes of oscillation: A rope or a string tied is tied down at both ends, like in a guitar or violin. We can make waves travel back and forth on it with a speed v in a complicated pattern, with different parts of the string moving in different ways. However, when the frequency is right, the string can vibrate in a standing wave, called a normal mode, in which every point of the string oscillates at the same special frequency. There are many normal modes, each with its own frequency. The first few normal modes of a string are shown below. λ/2 = L or λ 1 =2L L f 1 2λ/2 = L or λ 2 =L 2f 1 3λ/2 = L or λ 3 =2L/3 3f 1 With a wave speed v, f 1 = 2L v because λ= 2L There are many more normal modes each with a frequency nf 1 where n=1, 2, 3, 4,... Normal modes or standing waves on a string of length L tied at both ends 3

4 The frequencies of oscillation are multiples of the first harmonic f 1, 2f 1, 3f 1, 4f 1,... Other vibrating systems may have a different family of harmonics, such as only the odd ones, f 1, 3f 1, 5f 1,... This idea that only certain special characteristic frequencies are possible, is fundamental in music. But it is also fundamental in Quantum Mechanics. Electromagnetic waves: Electromagnetic waves travel in all directions, so their wave fronts are spheres. Their amplitude represents the strength of the electric field. Their intensity, which is proportional to the square of the amplitude, falls as 1/r 2 because the energy is spread over a sphere of radius r. In a vacuum all electromagnetic waves, radio, microwave, light, X-rays, etc., all travel with the speed of light c = km/s, which is actually the fastest speed possible in the universe. An electromagnetic wave consists of oscillating electric fields transverse to the direction of propagation, shown in red in the figure below, together with magnetic fields, shown in blue, that are also transverse and perpendicular to the electric fields. We say that such a wave is vertically polarized. An electric dipole antenna is shown in the figure below. The antenna radiates strongly in a direction perpendicular to the antenna and very weakly in the direction along the antenna wires. A vertical electric dipole antenna produces a vertically 4

5 polarized wave as shown above. The horizontal dipole antenna produces a horizontally polarized wave. Electric dipole antenna direction of strong wave perpendicular to antenna wire oscillating voltage zero radiation in this direction parallel to antenna zero radiation in this direction prallel to antenna λ/2 Polarization of Light: Visible light is emitted by atoms. In ordinary light sources the electric field changes direction randomly, because the atoms radiate independently of each other, and we say the light is unpolarized. This is why our eyes have not evolved to detect the polarization of light. However reflected light, and blue sky light are partially polarized. Polarized sunglasses shown below have a polarizing filter that only allows vertically polarized light to pass through. vertically polarized light passes through and enters eyes pass axis polarizing filter absorption axis horizontally polarized light is absorbed and does not pass vertical incoming E field vertical E field passed horizontal incoming E field polarizing filter horizontal E field absorbed and does not pass pass axis 5

6 They are useful because they tend to eliminate the glare produced by light reflected from horizontal surfaces. If a vertical filter is put in front of a horizontal filter, then no light passes through at all. The polarization of light is used in a clever way to produce the bright and dark pixels in LCD TV monitors. AM and FM waves: Audio frequencies for the normal human ear lie in the range from 20 Hz to 20 KHz. Above 20 khz it is ultrasound. However, the frequencies of the electromagnetic waves used for radio and TV transmission are much higher than audio frequencies. This is done because one wants an antenna of reasonable size. The length of a good dipole antenna to generate waves of wavelength λ should be roughly λ/2. But λ = c/f, where the speed of light is very large, c = km/s. Then if we tried to emit electromagnetic waves with f = 20, 000 Hz, the wavelength would be about 15 km, so it would require a huge antenna about 7.5 km long. To avoid this AM radio uses high frequency carrier waves with wavelengths in the range of 100 meters, while FM and TV use higher frequency carrier waves with wavelengths in the range of 1 meter. In the figure below the carrier wave has the frequency of the radio station. In order to transmit the music, or other audio signal, the carrier wave is modulated. In AM modulation the amplitude of the carrier wave is multiplied (modulated) by the low frequency audio signal as shown below. 6

7 In the case of FM modulation the amplitude and frequency of the signal is used to vary (modulate) the frequency of the carrier wave. The receiver filters out the carrier, and amplifies the audio signal, which is sent to a speaker. Interference and Coherent and Incoherent Sources: Two sources of waves are coherent if they have the same frequency or wave shape, and incoherent if they have different frequencies or wave shapes. Coherent waves can reinforce or cancel each other. In constructive interference the wave received is stronger because the crests (or peaks) of the two waves arrive together in a regular (coherent) manner. Then the amplitude of the resultant wave may be twice that of only one wave, and the intensity will be 4 times rather than 2 times the intensity of one wave. In destructive interference the wave received is canceled out because the crests (or peaks) of one wave arrive together with the troughs of the other wave in a regular (coherent) manner. Then the amplitude and intensity of the resultant wave will be zero, as shown below. 7

8 coherent sources have the same frequency or wave shape coherent sources, constructive interference = coherent sources, destructive interference = incoherent sources: no interference incoherent sources have different frequencies or wave shapes Two sources of water ripples, or two speakers driven with exactly the same signal will interfere constructively or destructively at a point in space depending on the different distances from the point to the two sources as seen below. If one source produces a wave of a different frequency, or of a completely different shape, the crests and troughs of the two waves reinforce at one moment but cancel at another moment in a complicated or random (incoherent) manner. Then the interference averages out, and the resultant intensity is merely 8

9 the sum of the two intensities. This is why two flash lights shining on a spot do not cancel out, but rather always give you more the sum of the intensities of each flashlight. Ordinary sound and light sources are incoherent with each other. Therefore the intensities of the sources add together. They are not canceled, nor doubled. Single Slit Diffraction: When waves of any kind pass through an opening they open out, or diffract, producing a pattern that depends on the size and shape of the opening. The figure below shows the diffraction pattern when water waves pass through an opening. One might have thought that the waves would be restricted to a beam the size of the opening, but instead they open out with maximae and minimae in different directions. The appearance of sharp shadows tells us that light travels in straight lines, and it is not obvious that light waves also diffract. What happens is that the amount of diffraction depends on the ratio λ/a of the wavelength divided by the width of the opening. If λ/a 1 the diffraction is negligible. This is the usual case with light because its wavelength is small compared to most openings, such as windows. However, with a narrow slit the diffraction is very similar to the pattern of water ripples. A laser beam, which is coherent across its face, its aimed at a single slit produces a 9

10 diffraction pattern projected on a screen as shown below. A laser beam aimed at an opaque steel ball produces a diffraction pattern projected on a screen as shown below. Surprisingly there is no obvious shadow, and there is the Poisson bright spot at the center of what one would have thought would be the shadow. This happens because the light is coherent all around the ball. Double Slit Diffraction and Interference: When water waves, or light waves, go through two slits, they diffract and interfere as if from two coherent sources as shown below for water ripples. When the laser beam falls on a double slit the slits act as coherent sources, and produce the pattern below. We get a complicated pattern. What you see is the double slit interference modulated by the single slit diffraction pattern. 10

11 Quantum Mechanics or Wave Mechanics: Quantum Mechanics shows that waves van behave like particles and particles can behave like waves. The modern view of quantum electromagnetic fields is that a field of frequency f has wave properties, but it can only change its energy when it interacts with matter (charged particles) in multiples of a basic quantum of energy we call a photon, or particle of light. The energy of a photon in light of frequency f is E = hf, where h = Joule second. Interestingly, this idea is due to Einstein, who never really believed in Quantum Mechanics. Phenomena like interference are explained using the wave properties of light, but it is a peculiarity of quantum mechanics that lets this happen even if there is only one photon in the field. Similarly particles like electrons are also entities in a quantum matter field, and they have wave properties. An electron moving with linear momentum p has properties like a wave of wavelength λ = h/p. Experiments analogous to the double slit can be done not only with light, but also with electrons, and other particles. 11

12 The magnitude (the strength) of the wave, usually called ψ, at a point is a measure of finding the particle at that point if one does an experiment to find where it is. Therefore quantum physics is probabilistic. Einstein objected to this and said something like God does not play dice. He proposed the Einstein-Podolsky-Rosen paradox as an objection. However, it turns out that the objection fails in experiments that show the truth of something called quantum entanglement as predicted by the Bell inequalities. An interesting article on this titled Is the Moon there when nobody looks? Reality and the quantum theory by the well known physicist David Mermin, appeared in the April 1985 issue of the magazine Physics Today. Some physicists say that Bell theorem is the most profound discovery of science. Others say that anybody who s not bothered by Bell s theorem has to have rocks in his head. 12