2. Waves and the Wave Equation

1 2. Waves and the Wave EquationWhat is a wave?Forward vs. backward propagating wavesThe one-dimensional wave equationPhase velocityReminders about complex numbersThe complex amplitude of a waveWhat is a wave?In the mathematical sense, a wave is any function that displace any function f(x)to the right, just change itsargument from x to x-x0,where x0is a positive we let x0= v t, where v is positive and t is time, then the displacement increases with increasing f(x-vt)represents a rightward, or forward, propagating , f(x+vt)represents a leftward, or backward, propagating is the velocity of the (x)f(x-1)f(x-2)f(x-3)The wave Equation in one dimension Later, we will derive the wave Equation from Maxwell s equations. Here it is, in its one-dimensional form for scalar ( , non-vector) functions, Equation determines the properties of most wave phenomena, not only light many real-world situations, the velocity of a wave depends on its amplitude, so v = v(f).

2 In this case, the solutions can be hard to , this is not the case for electromagnetic water waveair waveearth waveThe wave Equation is linear: The principle of Superposition has important consequences for light Waves . It means that light beams can pass through each other without altering each also means that Waves can constructively or destructively f1(x,t)and f2(x,t)are solutions to the wave Equation ,then their sum f1(x,t) + f2(x,t)is also a :and 22222 21212112 22222222221110vvvfffffff fxtxtxt 2221212222ffffxxx 2221212222ffffttt What if superposition wasn t true?That would mean that two Waves would interact with each other when passing through each other. This leads to some truly odd wave collisionswaves anti-crossingwaves spiraling around each otherThe solution to the one-dimensional wave equationThe wave Equation has the simple solution:If this is a solution to the Equation , it seems pretty it at all useful?

3 First, let s prove that it isa f (u)can be anytwice-differentiable function. ,fxtf x vt Proof that f(x vt)solves the wave equationWrite f(x vt)as f(u), where u = x vt. So and 1ux vut ffxu vfftu 22222vfftu 2222ffxu So and QED22 22222222211v0vvff ffxt uu Substituting into the wave Equation :ffuxux ffutut Now, use the chain rule:The 1D wave Equation for light waves22220 EExt where:E(x,t)is the electric field is the magnetic permeability is the dielectric permittivityThis is a linear, second-order, homogeneous differential useful thing to know about such equations:The most general solution has two unknown constants, which cannot be determined without some additional information about the problem ( , initial conditions or boundary conditions).And:We might expect that oscillatory solutions (sines and cosines) will be very relevant for light wave Equation : some solutionsWe showed that any twice-differentiable function can be a solution, as long as z and t appear in the right combination.

4 ,53 Eztz t So this is a solution:zE field amplitudeE(z) at t = 0E(z) at a later timeBut these are not really very useful solutions. 6,ztEzt e And this is a solution:zE field amplitudeE(z) at t = 0E(z) at a later time(,)cos[( c)]sin[( c)]Ezt B kzt C kzt (c)kzk t ( , )cos()sin()E z tBkztCkzt 1ck where:Note: this is the Greek letter omega, . Do notget it confused with w .It is more useful to use cosine- and sine-wave solutions:A more useful form for the solution( , )cos()sin()Ezt Bkzt Ckzt For simplicity, we ll just use the forward-propagating wave for now, so no . Now we can rewrite this in another form, using a trigonometric identity:cos(x y) = cos(x) cos(y) + sin(x) sin(y)With this identity, our solution becomes:E(z,t) =A cos[(kz t) ]Even more useful form for the solution( , )cos cos()sin sin()E z tAkztAkzt Thus our solution to the wave Equation becomes:Acos( ) = Band Asin( ) = CInstead of the unknown constants Band C, let s use two different unknown constants, Aand.

5 We define them so that:Definitions: Amplitude and Absolute phaseAbsolute phase = 0position, zat t = 0 Absolute phase = 2 /3 AThis is a common way of writing the solution to the wave Equation : ,cosEzt Akz t A= Amplitude(we will see that this is related to the wave s energy) = Absolute phase(or initial phase: the phase when z = t = 0)Clarification: What does this graph mean?Just as an illustration, here is the plot again for Absolute phase = 0position, zat t = 0 What are we plotting here? Be sure you understand , zat t = 0 Amplitude ofE field vectorThis picture contains no information about which way the E field is pointing! Only about the length of the vector at any point on the z axis, not about its axisA function of both z andt ,cosEzt Akz t Note:if you take a snapshot at any instant of time, the magnitude of E oscillates as a function of :if you sit at any location z, the magnitude of E oscillates as a function of quantities: Temporal quantities:xWavelength wave vector: k= 2 / wave number: = 1/ =k Temporal quantities:tPeriod angular frequency: = 2 / frequency: f= 1/ = For a given time, t0:For a given position, x0:The VelocityHow fast is the wave traveling?

6 Velocity is a reference distancedivided by a reference terms of the k-vector, k= 2 / , and the angular frequency, = 2 f, this is:c= / kThe velocity is the wavelength / period:c= / = fz The wave moves one wavelength, , in one period, .Do you need to memorize the value of the speed of light in empty space? : c0= 3 108meters / secondNote: this is the only constant I expect you to Phase of a WaveThe phase, , is everything inside the (x,t) = Acos( ), where = kx t Don t confuse the phase with the absolute phase (or initial phase ).The angular frequency and wave vector can be expressed as derivatives of the phase: = / tk= / xComplex numbersConsider 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 :1 jSo instead of using an ordered pair, we write:P = x + jyThe controversy: ivsjIn this class:We strive to be engineers and use (don t we won t see current too often.)

7 In physics: 1 iand j= current densityIn engineering: 1 jand i= currentEuler's Formulacossin jej so the point,P= A cos( ) + jAsin( ), can be written: P= Aexp( j )Truly one of the most important equations in all the :A= Amplitude = PhaseReminder: |exp( j )| = 1, for any (real) value of . Waves using complex numbersWe often write these expressions without the , Re, or + have seen that the electric field of a light wave can be written:E(x,t)= A cos(kx t )Since exp(j ) = cos( ) + jsin( ), E(x,t)can also be written:orwhere "+ " means "plus the complex conjugate of everything before the plus sign." 1, kxtcc ,ReexpExtAjkx t Waves using complex amplitudes ,expexp,p()ex Ext AjkxtExtjkxtAj How do you know if E0is real or complex?Sometimes people use the "~", but not always assume it's complex unless told otherwise. 0,exp Ext Ejkx t 0exp() (note the " ~ " symbol) EAj The resulting "complex amplitude" is: We can let the amplitude be complex:where we've separated theconstant stufffrom the rapidly changing not depend on x or complex amplitude of a waveRemember, nothing measurable ever contains numbers are merely a useful bookkeeping tool for tracking the phase of a quantity.

8 They don t appear in measurements. 0,exp Ext Ejkx t 0exp EAj whereThe amplitude of an electric field like this one is a quantity that (in principle) we can measure. It has units: volts/meter. This measurable quantity is never evercomplex. In principle, we can also measure the phase of this quantity (inradians). This value is also alwaysa real EA0 E