The wave equation and wave packets

Transcription

1 The wave equation and wave packets P. Nelson PHYS240/250 Spring Maxwell s Equations In first-year physics you learned about Maxwell s equations for electrodynamics in vacuum. These equations were probably expressed in terms of integrals, as follows: Gauss s Laws: Faraday s Law: Ampère s Law: B ds =0 E ds =q/ɛ 0 E d l = d B ds dt [ B d d l =µ 0 J + ɛ 0 dt ] E ds (1.1a) (1.1b) (1.1c) (1.1d) Here E is the electric field and B is the magnetic field. In the first two formulas, d S is an area integral over some closed surface that encloses charge q. In the second two, d S is an area integral over some open surface, d l is a line integral around the edge of that surface, and J is the electric current crossing it. The constant µ 0 tells how much magnetic field we get from a given current, while ɛ 0 tells how much electric field we get from a given charge. Please review any first-year text to make sure you remember what the symbols mean, where these equations came from, and what they mean in simple situations. Key remarks: First, eqs. (1.1) become linear equations if we set q =0and J =0. Thus if I give a solution, and you give another one, we can add them to get a third solution. Second, the Maxwell equations can be used to predict the future from the past: If you tell me the charges, currents, and fields at time t, Ican tell you the fields at time t +dt. It turns out that, although (1.1a d) are very useful in some situations, another formulation in terms of differential equations is best in other situations. It would be a long digression to get there unless you know some vector calculus (Stokes s theorem), but it s easy to get a special case. My goal is to arrive at the wave equation. 2. Waves First let s take the case of empty space, no currents nor charges: q = J =0. Second let s try to find solutions of a special form, namely E(x, y, z, t) =ˆxe(z,t), B(x, y, z, t) =ŷb(z,t). (2.1) 1

2 Here ˆx, ŷ are unit vectors and e, b are two ordinary functions of z and time. In words, (2.1) describes a transverse plane wave moving along ẑ: It moves along ẑ because E and B depend only on z,t, not on x, y. It s transverse because E, B are both perpendicular to the direction of motion. It s plane because E and B are constant over the planes {z =const,t=const}. I just want to show that some functions of the form (2.1) do solve (1.1, ) and see what the functions e, b may be. It will turn out that they must satisfy a certain differential equation. First check Gauss s laws: We examine eqn. (1.1b), where the surface is a tiny cubic box in space. Because E points along ˆx, the left side of the equation only gets contributions from the two faces of this cube lying parallel to the yz plane. But the outward-pointing normals d S to these faces point in opposite directions. Since E is independent of x, the two contributions cancel! For a volume which isn t a tiny cube, break it down into a bunch of tiny cubes and do the same argument for every one. In short, we just found that our proposed solution (2.1) does satisfy the Gauss law (1.1b). The same argument works for the magnetic Gauss law (1.1a). Next Faraday s law (1.1c): This time we need to choose a tiny square area. Let s take it in the xz plane (please show that squares in the xy or yz planes don t give anything interesting): Then we get no contribution to the line integral from sides (a), (c). The integrals are easy because the integrand is just a constant. Start with the left side of (1.1c): E d l =dxe(z +dz,t) dxe(z,t) =dxdz e z. (It s customary to replace the derivative symbol d by the curly-d, or, toemphasize that the functions e, b are functions of two variables z, t, and that e z for example means we wiggle z holding t fixed.) Faraday s law says this equals the right side of (1.1c), namely t (dxdzb(z,t)). Notice that both sides are proportional to dxdz, sowecan cancel these factors to get e z = b t. (2.2) That s a differential equation, which is what we wanted. But it s 2

3 oneequation in two unknown functions e, b. Before we can solve it, we must find a second equation. Finally Ampère s law (1.1d): Now it turns out (try it out) that the only interesting case is for a small square in the yz plane: Again we get contributions only from sides (b), (d): B dl =dyb(z,t) dyb(z +dz,t) = dxdz b z. Setting this equal to the other side, µ 0 ɛ 0 t (dxdze(z,t)), gives b z = µ 0 ɛ e 0 t. Let s abbreviate (µ 0 ɛ 0 ) 1/2 as c. Combining the above results we get the Wave Equation c 2 2 e z 2 2 e t 2 =0. Why call it that? Well, all we need to do to get a solution is to choose e(z,t) tobeany function of z ± ct. Think about it for a moment: This means that any waveform moving upwards or downwards at constant velocity c is a solution to Maxwell s equations.* These waves don t change their shape as they move; a sharp pulse stays sharp, etc. If a sharp pulse instead tended to smear out we d say the equation had dispersion; the wave equation has none. Maxwell realized that the amazing thing is that µ 0 and ɛ 0 can be measured using magnets, charged balls, and so on; they have no obvious relation at all to light. And yet, this combination c turns out to be m/sec, which is the speed of light! Maxwell could not help guessing that light was exactly a kind of electromagnetic wave. Later Hertz * There s nothing special about up or down; it could move in any direction, but I selected the z axis for illustration. That was in vacuum. In a medium, like glass, the speed of light is less than c, and depends on wavelength; there is dispersion. That s why a prism can separate light into its pure components. Later we ll see a different way to separate light using diffraction. 3

4 gave even more weight to this guess by creating radio waves from an apparatus which was obviously electromagnetic, and then noticing that they displayed all the behavior of light (polarization, diffraction, refraction,...).* But what really made people sit up and take notice was when Marconi (and others) took these abstruse, theoretical ideas and converted them into life-saving technology: the ability to communicate with ships at sea. Everybody could understand the importance of radio. Einstein thought long and hard about the wave equation, and so will we. One key comment: there s no dispersion because every wave travels at speed c. This is only possible because Maxwell s equations contain some universal constant of nature (µ 0 ɛ 0 ) 1 with dimensions of velocity. By the way, a very similar-looking equation governs the passage of sound through the air. But there s a difference, as we ll see. Finally, the wave equation has some similarities to and differences from the Schrödinger equation. For instance, while the S.E. also has wave solutions, it is not free from dispersion. Try this: Schrödinger s equation for an electron contains only the constants m e, h. Show that this means so there s no way it could give dispersionless solutions!! Try this: So far, we ve assumed that the charge and current is zero (empty space). In that case, one possible solution is E = B =0! To understand generation of waves, now suppose that there is a thin sheet of current in the plane {z =0}, and that this current is oscillating back and forth in the ˆx direction. Again look for a solution of the modified wave equation. Ans: e c 2 ë = µ 0 J x δ(z) where J x = A cos(ωt) is current per length. Integrating across the singularity sets a boundary condition, which we can meet by e(z,t) =B cos(ω(t x /c)). Substituting we find B in terms of the current strength A. 3. Energy transport Waves can carry energy from place to place. If I shake one end of a rope, you can use it to drive a little motor at the other end and get useful work done. Similarly, a radio station pumps energy out into the air which never comes back; some proceeds into outer space, while a tiny fraction lands on my antenna at home and does some work shaking the electrons there. The right way to think of this is to say that when we shake a rope a certain power (energy per time) goes into the motion of the rope and gets turned into a wave, which then moves away. In three dimensions it s a little more subtle. Here the wave spreads out in all directions in space, so it has to get weaker. At any point the wave carries a power per area, or * For the record, David Edward Hughes did some of the same experiments several years before Hertz, and correctly interpreted them. Sometimes the world isn t ready for a good idea. 4

5 (energy)/(time area), called the energy flux. Ifwesurround a radio transmitter by a surface, and measure the energy flux through each unit of surface, and integrate over the whole surface, we get the total power leaving the antenna. Whatever the surface we choose, this total power must equal the power going into the transitter (conservation of energy). You learned in first-year physics that the energy flux in an electromagnetic wave is µ 0 1 E B, the Poynting vector. Try this: (a) Consider a wave described by (2.1) with e(z,t) = A sin(k(z ct)) for some constant k. Using (2.1), show that the magnitude of the Poynting vector (rate of energy transport per area) can be written as (µ 0 c) 1 ( E) 2. Actually, however, we usually want the 1 time average of this oscillating quantity; show it s 2µ 0 c ( E max ) 2. (b) Now find the direction of the Poynting vector. Repeat for the wave with e(z,t) = A sin(k(z ct)) and comment. The important thing to remember is that any wave transports energy at a rate proportional to the square of the amplitude. This applies to water waves, sound waves, waves in violin strings, and (as we just saw) light. 4. Sines and Cosines You may have gotten the impression in earlier courses that light waves had to be sinusoidal. We just saw this isn t the case at all. Sine waves are a special situation: monochromatic light, with color determined by the wavelength of the sine wave. The wave equation is linear because Maxwell s equations were.* This turns out to be a big help. Here I want to convince you that a general waveform can be thought of as the sum of a bunch of simpler ones, namely sine-type waves. You ll learn the deep theory in some math class. As usual I just want to give an illustration which I hope will make the point. We can write a sinusoidal wave as f(x) =sin(kx), or more generally sin(kx+φ). Here k is called the wavenumber and φ is called the phase shift. I m claiming that we can synthesize any waveform out of a bunch of pure sine waves, exactly as a music synthesizer creates a complicated musical tone out of several pure tones. Turning it around, we can analyze a complex waveform into its simple bits, just as Newton split white light into pure colors. Once we know this, we also know the time evolution: just decompose the initial wave into sine waves and let each wave evolve forward in time. The linearity of the wave equation then tells us that the solution at a later time can be found by recombining all these traveling waves. * Ultimately we ll be more interested in the Schrodinger equation it, too, is linear. 5

6 As an example, suppose that we want a pulse of sine wave, like a short note played on an organ. That is, we want a tone of some short wavelength, modulated by an overall envelope function which is zero for a long time, then rises and falls, then is zero after that. To get started, let s try putting together two sine waves of wavenumber k =19and 21. Let s get our assistant to help. I wrote plot({sin(19*x),sin(21*x)},x=-5.. 5, numpoints=300, resolution=600,color=black); to get We see the phenomenon of beating: the waves start out in phase, get out of phase, then back in, etc. If now I add these two waves I get That s a bit like what I want, but the envelope function isn t a single pulse but rather a series of pulses. Try this: Explain the above picture using properties of complex exponentials. Show that the tone in each pulse has wavenumber 20 (i.e. wavelength 2π/20) while the envelope function has wavenumber 1 (wavelength 2π). To do better we start combining more waves, but always with wavenumber close to 20. The trick is to get just the right amount of each wave, i.e. combine them with the right weight. We could try to suppress every alternate pulse in the above picture. By the beat idea, we could try to do it by introducing a tone right at wavenumber 20. Some fooling around leads us to 6

7 which is getting closer. Isn t there some more systematic way? Yes, but I m not going to prove it. If you stick together a bunch of waves at various wavenumbers k and give each the weight weight = e (k 20)2 l 2 /2, (4.1) you get a pulse (or wavepacket) oflength about equal to l, and in the pulse a tone of wavenumber 20. What this means is that we take the various waves sin kx, multiply each by the weight given above for the given k, and add them all up. Let s try just sticking together twelve waves at k = 17.25,...,22.75 according to the recipe (4.1). For example, I tried l =1. Ityped epsilon:=.5: N:=6: f:=x -> sum(exp(-(epsilon*(i-.5))ˆ2/2)*(sin(x*(20+(i-.5)*epsilon))+sin(x*(20- (i-.5)*epsilon))),i=1..n); Then I graphed f to get 7

8 Even though I didn t justify the recipe (4.1) rigorously, the previous examples make one thing clear: to get a pulse of length l, combine pure waves in a range of wavenumbers of width about l 1. We already saw this in our two-wave model: to get longer beats, choose two waves closer together. Similarly, the weight function (4.1) drops off when k differs from 20 by more than about l 1. Again: The shorter the pulse in space, the less well-defined the wavenumber; the longer the pulse, the more sharply-defined the wavenumber. That is, there s a reciprocal relation between duration and purity of musical tone. Some day you ll dress this easy remark up in French clothes and call it Fourier analysis, but to remember it, always think first about the combination of two waves: If the wavenumber k is sharply defined, we have to go longer before the two waves get 180 out of phase. So the beats are of longer duration. Just for fun: Try this: > with(plots): > epsilon:=.4: N:=10: f:= (x,t) ->sum( exp(-(epsilon*(i-.5))ˆ2/32) *( sin(x*(20+(i-.5)*epsilon)+t*((20+(i-.5)*epsilon)ˆ2)) +sin(x*(20-(i-.5)*epsilon)+t*((20-(i-.5)*epsilon)ˆ2)) ),i=1..n); > animate(f(x,t),x=-3.. 3,t= ,numpoints=200, frames=35); We ll discuss the meaning of this pretty movie later on. 8