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Lecture 4 Plane Waves 3D Differential Wave Equation Spherical Waves Cylindrical Waves

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Lecture 4 Plane Waves 3D Differential Wave Equation Spherical Waves Cylindrical Waves Lecture 4 Chapter 2 Wave Motion Plane waves 3D Differential wave equation Spherical waves Cylindrical waves 3-D waves: plane waves (simplest 3-D waves) All the surfaces of constant phase of disturbance form parallel planes that are perpendicular to the propagation direction 3-D waves: plane waves (simplest 3-D waves) All the surfaces of constant phase of disturbance form parallel planes that are perpendicular to the propagation direction An equation of plane that is ˆ ˆ ˆ perpendicular to k kx i k y j kzk Unit vectors k r const a All possible coordinates of vector r are on a plane k Can construct a set of planes over which varies in space harmonically: r Asink r or r Acosk r or r Aeikr Plane waves r sink r The spatially repetitive nature can be expressed as: k r r k In exponential form: r Aeikr Aeikr k / k Aeikr eik For that to be true: ei2 1 k 2 2 k Vector k is called propagation vector Plane waves: equation r Aeik r This is snap-shot in time, no time dependence To make it move need to add time dependence the same way as for one-dimensional wave: r,t Aeik r t Plane wave equation Plane wave: propagation velocity Can simplify to 1-D case assuming that wave propagates along x: ˆ r || i ikxt r,t Aeikr t r,t Ae We have shown that for 1-D wave phase velocity is: That is true for any direction of k v + propagate with k k - propagate opposite to k More general case: see page 26 Example: two plane waves Same wavelength: k1= k2=k=2/, Write equations for both waves. Solution: Aeikr t Same speed v: 1=2==kv Dot product: k r kx x k y y kz z Wave 1: k1 r k1z Wave 2: k2 r k2 sin y k2 cos z direction ik1zt ik2 y sin z cos t 1 A1e 2 A2e 1 A1 coskz t 2 A2 cosky sin z cos t Note: in overlapping region = 1 + 2 Plane waves: Cartesian coordinates ik r t r,t Ae k r kx x k y y kz z Wave eq-ns in Cartesian coordinates: x, y, z,t Aeikx xk y ykz zt x, y, z,t Aeik xyz t , , - direction cosines of k 2 2 2 k kx k y kz 2 2 2 1 Importance of plane waves: • easy to generate using any harmonic generator • any 3D wave can be expressed as superposition of plane waves Three dimensional differential wave equation Taking second derivatives for x, y, z,t Aeik xyz t can derive the following: 2 2 2 2 k 2 x2 t2 + 2 2 1 2 2 k 2 t2 y2 + 2 2 2 combine and use: v k k z2 2 2 2 k 2 x2 y2 z2 3-D differential wave equation 2 2 2 1 2 x2 y2 z2 v2 t2 Three dimensional differential wave equation 2 2 2 1 2 Alternative expression x2 y2 z2 v2 t2 Use Laplacian operator: 2 2 2 2 2 1 2 2 2 x2 y2 z2 v t Using =kv, we can rewrite x, y, z,t Aeik xyz t x, y, z,t Aeik xyzvt function of x y z vt It can be shown, that: x, y, z,t f x y z vt f, g are plane-wave solutions of the diff. eq- n, provided that are twice differentiable. x, y, z,t g x y z vt Not necessarily harmonic! In more general form, the combination is also a solution: x, y, z,t C1 f r k / k vt C2 gr k / k vt Example Given expression: x , t ax bt c 2 , where a>0, b>0 Does it correspond to a traveling wave? What is its speed? Solution: 1. Function must be twice differentiable 2ax bt c a 2ax bt c b x t 2 2 2a2 2b2 x2 t 2 2. Speed: 2 2 2 1 2 x2 y2 z2 v2 t2 1 b 2a2 2b2 v v2 a Direction: negative x direction Example Given expression x , t ax 2 bt ,where a>0, b>0: Does it correspond to a traveling wave? What is its speed? Solution: 1. Function must be twice differentiable 2x3 b x t 2 2 4 6ax 2 0 x2 t 2. Wave equation: 2 2 2 1 2 x2 y2 z2 v2 t2 6ax4 0 Is not solution of wave equation! This is not a wave traveling at constant speed! Spherical waves Spherical waves originate from a point 2-D concentric water waves source and propagate at constant speed in all directions: waveforms are concentric spheres. Isotropic source - generates waves in all directions. spherical wave Symmetry: introduce spherical coordinates x r sin cos y r sin sin z r cos Symmetry: the phase of wave should only depend on r, not on angles: r r, , r 1 2 2 Spherical waves v2 t2 Since depends only on r: 2 1 2 r r2 r r 1 2 evaluates to the same 2 r r r2 Wave equation: x r sin cos y r sin sin 1 2 1 2 r ×r z r cos r r2 v2 t 2 2 1 2 2 r 2 1 2 r r r r r 2 2 2 1 r v t sin r2 sin 1 2 r2 sin2 2 Spherical waves 2 1 2 r r r2 v2 t 2 This is just 1-D wave equation In analogy, the solution is: r r,t f r vt f r vt - propagates outwards (diverging) r,t r + propagates inward (converging) Note: solution blows up at r=0 In general, superposition works too: f r vt gr vt r,t C C 1 r 2 r Harmonic Spherical waves f r vt r,t r In analogy with 1D wave: Harmonic spherical wave A r,t coskr vt r A ik rvt A - source strength r,t e r Single propagating Constant phase at any given time: kr=const pulse Amplitude decreases with r A Spherical harmonic waves A r,t coskr vt r Decreasing amplitude makes sense: Waves can transport energy (even though matter does not move) The area over which the energy is distributed as wave moves outwards increases Amplitude of the wave must drop! Note: spherical waves far from source approach plane waves: Cylindrical waves Wavefronts form concentric cylinders of infinite length Symmetry: work in cylindrical coordinates x r cos r r, , z r y r sin z z 1 1 2 r r r r v2 t 2 It similar to Bessel’s eq-n. At larger r the solution can be approximated: Harmonic cylindrical wave A r,t coskr vt 2 1 r r r r r A 2 2 ik rvt 1 r,t e r r2 2 z2 Cylindrical waves Harmonic cylindrical wave A r,t coskr vt r A r,t eik rvt r Can create a long wave source by cutting a slit and directing plane waves at it: emerging waves would be cylindrical. Lecture 5 Chapter 3 Electromagnetic theory, Photons. and Light Basic laws of electromagnetic theory Maxwell’s equations Electromagnetic waves Polarization of EM waves Energy and momentum Basic laws of electromagnetic theory Electric field Coulomb force law: 1 Q1Q2 F F 2 40 r electric permittivity of free space Q1 Q2 Black box F F F EQ1 Interaction occurs via electric field Electric field can exist even when charge 1 Q2 E 2 rˆ disappears (annihilation in black box) 40 r Basic laws of electromagnetic theory Magnetic field Moving charges create magnetic field permeability of free space The Biot-Savart law for qv rˆ B 0 moving charge 4 r2 Magnetic field interacts with moving charges: Fmagnetic qv B Charges interact with both fields: F qE qv B (Lorentz force) Basic laws of electromagnetic theory Gauss’s Law: electric Karl Friedrich Gauss (1777-1855) Electric field flux from an enclosed volume is proportional to the amount of charge inside 1 E q 0 If there are no charges (no 1 sources of E field), the flux E dS q S is zero: 0 E dS 0 S More general form: 1 E dS dV S V 0 Charge density Basic laws of electromagnetic theory Gauss’s Law: magnetic Magnetic field flux from an enclosed volume is zero (no magnetic monopoles) M 0 B dS 0 S Basic laws of electromagnetic theory Faraday’s Induction Law 1822: Michael Faraday Changing magnetic field can result in variable electric field d emf M dt normal to area Formal d area E dl B dS version C dt A dS nˆdA Changing current in the d B dS B nˆdA solenoid produces changing M magnetic field B. Changing d M BdAcos magnetic field flux creates angle between B and electric field in the outer wire. normal to the area dA Basic laws of electromagnetic theory Ampère’s Circuital Law 1826: (Memoir on the Mathematical Theory of Electrodynamic Phenomena, Uniquely Deduced from Experience) A wire with current creates magnetic field around it All the currents in the universe contribute to B but only ones inside the path result in nonzero path integral Ampere’s law B dl 0 Iinside_ path C Incomplete! B dl 0 J dS C A Current density Basic laws of electromagnetic theory Ampère’s-Maxwell’s Law Maxwell considered all known laws and noticed asymmetry: Gauss’s B dS 0 S 1 Gauss’s E dS q S 0 d Changing magnetic field Faraday’s E dl B dS C dt A leads to changing electric field Ampère’s B dl 0 J dS C A No similar term here Hypothesis: changing electric field leads to variable magnetic field Basic laws of electromagnetic theory Ampère’s law Ampère’s-Maxwell’s Law B dl 0 J dS C A The B will depend on area: B dl 0 J dS 0i C A 1 B dl 0 J dS 0 C A 2 Workaround: Include term that takes into account changing electric field flux in area A : 2 E B dl 0 J 0 dS Ampère’s-Maxwell’s Law: C A t displacement current density Maxwell equations Gauss’s B dS 0 S 1 In vacuum Gauss’s E dS q S (free space) 0 d Faraday’s E dl B dS C dt A Ampère- E B dl 0 J 0 dS C A Maxwell’s t + Lorentz force: F qE qv B fields are defined through interaction with charges Inside the media electric and magnetic fields are scaled.
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