Waves, the Wave Equation, and Phase Velocity. We ll start with optics. The onedimensional wave equation. What is a wave? Optional optics texts: f(x)


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1 We ll start with optics Optional optics texts: Waves, the Wave Equation, and Phase Velocity What is a wave? f(x) f(x) f(x) f(x3) Eugene Hecht, Optics, 4th ed. J.F. James, A Student's Guide to Fourier Transforms Forward [f(xvt)] and backward [f(x+vt)] propagating waves The onedimensional wave equation Harmonic waves Optics played a major role in all the physics revolutions of the 0th century, so we ll do some. Wavelength, frequency, period, etc. And waves and the Fourier transform play major roles in all of science, so we ll do that, too. Complex numbers A wave is anything that moves. If we let a = v t, where v is positive and t is time, then the displacement will increase with time. v will be the velocity of the wave. 3 x Plane waves and laser beams We ll derive the wave equation from Maxwell s equations. Here it is in its onedimensional form for scalar (i.e., nonvector) functions, f: f(x) f(x) f(x) f(x3) f x So f(x  v t) represents a rightward, or forward, propagating wave. Similarly, f(x + v t) represents a leftward, or backward, propagating wave. The onedimensional wave equation What is a wave? To displace any function f(x) to the right, just change its argument from x to xa, where a is a positive number. 0 Phase velocity 0 3 x " f v t = 0 Light waves (actually the electric fields of light waves) will be a solution to this equation. And v will be the velocity of light.
2 The solution to the onedimensional wave equation The wave equation has the simple solution: Proof that f (x ± vt) solves the wave equation u u Write f (x ± vt) as f (u), where u = x ± vt. So = and v x t = ± f f u Now, use the chain rule: = f f u = x u x t u t f ( x, t) = f ( x ± v t) f f So = f f f f = and = ± v x u x u t u f f t u = v Substituting into the wave equation: where f (u) can be any twicedifferentiable function. f f f " f # $ = $ v 0 % & = x v t u v ' u ( The D wave equation for light waves A simpler equation for a harmonic wave: E E x t # µ" = 0 where E is the light electric field Use the trigonometric identity: E(x,t) = A cos[(kx t) "] We ll use cosine and sinewave solutions: E( x, t) = B cos[ k( x ± v t)] + C sin[ k( x ± v t)] cos(z y) = cos(z) cos(y) + sin(z) sin(y) where z = k x t and y = " to obtain: or where: E( x, t) = B cos( kx ± t) + C sin( kx ± t) k kx ± ( kv) t = v = µ" The speed of light in vacuum, usually called c, is 3 x 0 0 cm/s. E(x,t) = A cos(kx t) cos(") + A sin(kx t) sin(") which is the same result as before, as long as: E( x, t) = B cos( kx " t) + C sin( kx " t) A cos(") = B and A sin(") = C For simplicity, we ll just use the forwardpropagating wave.
3 Definitions: Amplitude and Absolute phase E(x,t) = A cos[(k x t ) " ] A = Amplitude " = Absolute phase (or initial phase) Definitions Spatial quantities: " Temporal quantities: The Phase Velocity Human wave How fast is the wave traveling? Velocity is a reference distance divided by a reference time. The phase velocity is the wavelength / period: v = # / $ Since % = /$ : v = # v In terms of the kvector, k = " / #, and the angular frequency, = " / $, this is: v = / k A typical human wave has a phase velocity of about 0 seats per second.
4 The Phase of a Wave The phase is everything inside the cosine. E(x,t) = A cos(&), where & = k x t " Complex numbers Consider a point, P = (x,y), on a D Cartesian grid. In terms of the phase, And & = &(x,y,z,t) and is not a constant, like " = $& /$t k = $& /$x $& /$t v = $& /$x This formula is useful when the wave is really complicated. Let the xcoordinate be the real part and the ycoordinate the imaginary part of a complex number. So, instead of using an ordered pair, (x,y), we write: where i = () / P = x + i y = A cos(#) + i A sin(#) Euler's Formula exp(i&) = cos(&) + i sin(&) so the point, P = A cos(#) + i A sin(#), can be written: P = A exp(i&) Proof of Euler's Formula exp(i&) = cos(&) + i sin(&) 3 x x x Use Taylor Series: f ( x) = f (0) + f '(0) + f ''(0) + f '''(0) x x x x exp( x) = x x x x cos( x) = x x x x x sin( x) = where A = Amplitude & = Phase If we substitute x = i& into exp(x), then: 3 4 i i exp( i ) = + " " # $ # $ = % " i... 4 & + % " + 3 & ' ( ' ( = cos( ) + i sin( )
5 Complex number theorems If exp( i ) = cos( ) + i sin( ) exp( i ) = # exp( i / ) = i exp( i" ) = cos( ") # i sin( ") cos( ") = [ exp( i" ) + exp( # i" )] sin( ") = [ exp( i" ) # exp( # i" )] i A exp( i" ) $ A exp( i" ) = A A exp i( " + " ) A exp( i" ) / A exp( i" ) = A / A exp i( " #" ) More complex number theorems Any complex number, z, can be written: z = Re{ z } + i Im{ z } So Re{ z } = / ( z + z* ) and Im{ z } = /i ( z z* ) where z* is the complex conjugate of z ( i % i ) The "magnitude," z, of a complex number is: z = z z* = Re{ z } + Im{ z } To convert z into polar form, A exp(i&): A = Re{ z } + Im{ z } tan(#) = Im{ z } / Re{ z } We can also differentiate exp(ikx) as if the argument were real. Proof : d dx d exp( ikx) = ik exp( ikx) dx cos( kx) + i sin( kx) = k sin( kx) + ik cos( kx) " # = ik $ sin( kx ) + cos( kx ) & i % ' But / i = i, so : = ik i sin( kx) + cos( kx) Waves using complex numbers Recall that the electric field of a wave can be written: E(x,t) = A cos(kx t ") Since exp(i&) = cos(&) + i sin(&), E(x,t) can also be written: or E(x,t) = Re { A exp[i(kx t ")] } E(x,t) = / A exp[i(kx t ")] + c.c. where "+ c.c." means "plus the complex conjugate of everything before the plus sign." We often write these expressions without the, Re, or +c.c.
6 Waves using complex amplitudes We can let the amplitude be complex: where we've separated the constant stuff from the rapidly changing stuff. The resulting complex amplitude is: So: (, ) = exp ( # #" ) E x t A $ & i kx t %' { } (, ) = { Aexp ( i" )} exp ( x # ) E x t # $ & i k t %' E0 = Aexp( " i ) # (note the " ~ ") (, ) = exp ( " ) E x t E0 i kx t How do you know if E 0 is real or complex? As written, this entire field is complex Complex numbers simplify waves Adding waves of the same frequency, but different initial phase, yields a wave of the same frequency. This isn't so obvious using trigonometric functions, but it's easy with complex exponentials: Etot ( x, t) = E exp i( kx " t) + E exp i( kx " t) + E3 exp i( kx " t) = ( E + E + E3)exp i( kx " t) where all initial phases are lumped into E, E, and E 3. Sometimes people use the "~", but not always. So always assume it's complex. The 3D wave equation for the electric field and its solution A wave can propagate in any direction in space. So we must allow the space derivative to be 3D: µ " E " t # E $ = 0 E 0 exp[ i ( k " r # t )] " is called a plane wave. A plane wave s contours of maximum phase, called wavefronts or phasefronts, are planes. They extend over all space. or E E E E x y z t + + # µ" = 0 which has the solution: E( x, y, z, t) = E0 exp[ i( k " r # t)] " " where k ( kx, k y, kz) r ( x, y, z) and k r " k x + k y + k z x y z k k + k + k x y z Wavefronts are helpful for drawing pictures of interfering waves. A plane wave's wavefronts are equally spaced, a wavelength apart. They're perpendicular to the propagation direction. A wave's wavefronts sweep along at the speed of light. Usually, we just draw lines; it s easier.
7 Laser beams vs. Plane waves A plane wave has flat wavefronts throughout all space. It also has infinite energy. It doesn t exist in reality. A laser beam is more localized. We can approximate a laser beam as a plane wave vs. z times a Gaussian in x and y: " # x + y E( x, y, z, t) = E0 exp % $ exp[ i( kz $ t) ] ' ( w & z Localized wavefronts y x w Laser beam spot on wall
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