2. Waves, the Wave Equation, and Phase Velocity

1 Waves, the Wave Equation, and Phase VelocityWhat is a wave?Forward [f(x-vt)] and backward [f(x+vt)] propagating wavesThe one-dimensional wave equationWavelength, frequency, period, Velocity Complex numbers Plane waves and laser beams Boundary conditions Div, grad, curl, etc., and the 3D Wave equationf(x)f(x-3)f(x-2)f(x-1)x0 1 2 3 Source: Trebino, Georgia Te c hWhat is a wave?A wave is anything that displace any function f(x)to the right, just change its argument from xto x-a, where ais a positive we let a = vt, where vis positive and tis time, then the displacement will increase with represents a rightward, or forward, propagating , represents a leftward, or backward, propagating be the Velocity of the (x)f(x-3)f(x-2)f(x-1)x0 1 2 3f(x -vt)f(x + vt)The one-dimensional wave equation2222210vffxt = The one-dimensional wave equation for scalar ( , non-vector) functions, f.

2 Where vwill be the Velocity of the wave.( , )(v)fxt fx t= The wave equation has the simple solution:where f (u)can be any twice-differentiable that f (x vt)solves the wave equationWrite f(x vt)as f(u), where u = x vt. So and Now, use the chain rule:So and Substituting into the wave equation:1ux = vut = f fux ux = f fut ut = ffxu = vfftu = 22222vfftu = 2222ffxu = 22 2222 222 2211v0vvff ffxt uu = = The 1D wave equation for light wavesWe ll use cosine-and sine-wave solutions:or where.

3 22220 EExt = ( , )cos[ (v )]sin[ (v )]Ext Bkx tCkx t= + ( , )cos()sin()E x tBkxtCkxt = + 1vk ==( v)kxk t where Eis the light electric fieldThe speed of light in vacuum, usually called c , is 3 x1010 simpler equation for a harmonic wave:E(x,t) =A cos[(kx t) ]Use the trigonometric identity:cos(z y) = cos(z) cos(y) + sin(z) sin(y)where z = kx tand y = to obtain:E(x,t)= A cos(kx t) cos( ) + Asin(kx t) sin( )which is the same result as before, as long as:Acos( ) = Band Asin( ) = C( , )cos()sin()E x tBkxtCkxt = + For simplicity, we ll just use the forward-propagating : Amplitude and Absolute phaseE(x,t) = Acos[(k x t ) ]A= Amplitude = Absolute Phase (or initial Phase ) kxDefinitionsSpatial quantities: Temporal quantities:The Phase VelocityHow fast is the wave traveling?

4 Velocity is a reference distancedivided by a reference Phase Velocity is the wavelength / period: v= / Since = 1/ :In terms of the k-vector, k= 2 / , and the angular frequency, = 2 / , this is:v= v v= / k The Phase is everything inside the (x,t) = Acos( ), where = k x t = (x,y,z,t)and is not a constant, like !In terms of the Phase , = / tk= / xAnd / tv= / xThe Phase of a WaveThis formula is useful when the wave is really Narrows Bridge1. The animation shows the Tacoma Narrows Bridge shortly before its collapse. What is its frequency? HzD1 Hz2.

5 The distance between the bridge towers (nodes) was about 860 meters and there was also a midway node. What was the wavelength of the standing torsionalwave?A1720mB860 mC430mDThereis no way to What is theamplitude? mB4 mC8 mD16 mAnimation: numbersSo, instead of using an ordered pair, (x,y), we write:P = x + iy= Acos( ) + iAsin( )where i= (-1)1/2 Consider a point,P= (x,y), on a 2D Cartesian the x-coordinate be the real part and the y-coordinate the imaginary part of a complex 's Formulaexp(i ) = cos( ) + isin( )so the point,P= Acos( ) + i Asin( ), can be written: P= Aexp(i )whereA= Amplitude = PhaseProof of Euler's FormulaUse Taylor Series:2 34243exp( ) !

6 2!3!4! !4!1!3!cos( )sin( )iiiii =+ ++ = ++ + + =+23( )(0)'(0)''(0)'''(0) ..1!2 !3!xx xfx ffff=++ + +exp(i ) = cos( ) + i sin( )23424683579exp( ) !2!3!4!cos( ) !4!6!8!sin( )..1!3!5!7 !9 !xx x xxxxxxxxx x x xx=+++++= + ++= + ++If we substitutex = i into exp(x), then:Complex number theorems[][][][]1122121211 22121 2 exp( )1 exp(/ 2) exp(- ) cos( )sin( )1 cos( )exp( ) exp()21 sin( )exp( ) exp()2 exp()exp()exp () exp() /exp()/exp ()iiiiiiiiiiAiAiAAiAiA i AA i = == =+ = = += exp( ) cos( )sin( )ii =+If More complex number theoremsAny complex number,z, can be written.

7 Z= Re{ z} + iIm{z}SoRe{z} = 1/2 ( z + z*)andIm{ z} = 1/2i( z z* )wherez*is the complex conjugate of z( i i)The "magnitude," | z |, of a complex number is:| z |2= z z*= Re{ z }2+ Im{ z }2To convert zinto polar form, Aexp(i ): A2= Re{ z }2+ Im{ z }2tan( ) = Im{z } / Re{z }We can also differentiateexp(ikx)as if the argument were real.[][] exp()exp()cos( )sin( )sin( )cos( )1 sin( )cos( ) 1/ sin( ) cos( )dikxikikxdxdkxikxkkxikkxdxikkxkxiiiik ikxkx=+ = + = + ==+Proof :B u t, s o :Waves using complex numbersThe electric field of a light wave can be written:E(x,t)= A cos(kx t )Since exp(i ) = cos( ) + isin( ),E(x,t)can also be written:E(x,t) = Re { A exp[i(kx t )] }orE(x,t) = 1/2 A exp[i(kx t )]+ "+ " means "plus the complex conjugate of everything before the plus sign.

8 "We often write these expressions without the , Re, or + using complex amplitudesWe can let the amplitude be complex:where we've separated theconstant stufffrom the rapidly changing stuff. The resulting "complex amplitude" is: So:()()(){}(){},expex,ex(pp)E x tAi kxtE xtiktixA = = 0exp() EA i = (note the " ~ ")()()0,expE x tEi kxt = How do you know if E0is real or complex?Sometimes people use the "~", but not always assume it's written, this entire field is complex!Complex numbers simplify waves!This isn't so obvious using trigonometric functions, but it's easywith complex exponentials.

9 123123( , )exp ()exp ()exp () () exp ()totEx tEi kxtEi kxtEi kxtEEEi kxt = + + = ++ where all initial phases are lumped into E1, E2, and waves of the same frequency, but different initial Phase , yields a wave of the same called a plane plane wave's wave-fronts are equally spaced, a wavelength 're perpendicular to the propagation are helpful for drawing pictures of interfering wave's wave-fronts sweep along at the speed of plane wave s contours of maximum field, called wave-frontsor Phase -fronts, are planes. They extend over all [ ()]Ei kxt Usually, we just draw lines; it s waves in space: beamsA plane wavehas flat wave-fronts throughout all space.

10 It also has infinite doesn t exist in waves are more localized. We can approximate a realistic wave as a plane wave vs. ztimes a Gaussian in xand y:2220( , , , )exp[ ()exp]xyE x y z tEi kztw = + Laser beam spot on wallwxyLocalized wave-frontszxexp(-x2)Localized waves in time: pulsesIf we can localize the beam in space by multiplying by a Gaussian in xand y, we can also localize it in time by multiplying by a Gaussian in ( , , , )expexp[ (exp)]xyE x y z tEi kzttw + = tEThis is the equation for a laser (-t2)Longitudinal vs. Transverse wavesMotion is along the direction of propagation longitudinal polarizationMotion is transverse to the direction of propagation transverse polarizationSpace has 3 dimensions, of which 2 are transverse to the propagation direction, so there are 2 transverse waves in addition to the potential longitudinal direction of the wave s variations is called its :Longitudinal:Vector fieldsLight is a 3D vector 3D vector field assigns a 3D vector ( , an arrow having both direction and length) to each point in 3D space.