Waves, the Wave Equation, and Phase Velocity

Transcription

1 Waves, the Wave Equation, and Phase Velocity

2 What is a wave? The one-dimensional wave equation Wavelength, frequency, period, etc. Phase velocity Complex numbers and exponentials Plane waves, laser beams, and pulses

3 What is a wave? A wave is anything that moves. To displace any function f(x) to the right, just change its argument from x to x- a, where a is a positive number. f(x) f(x-2) f(x-1) f(x-3) If we let a = v t, where v is positive and t is time, then the displacement will increase with time. So f(x - v t) represents a rightward, or forward, propagating wave. Similarly, f(x + v t) represents a leftward, or backward, propagating wave. v will be the velocity of the wave x

4 The One-Dimensional Wave Equation and Its Solution Here it is in its one-dimensional form for scalar (i.e., non-vector) functions, f: f x 1 f - v t where v is the velocity of the wave Light waves (specifically, the electric fields of light waves) will be a solution to this equation. And v will be the velocity of light. The wave equation has the simple solution: f ( x, t) f ( x v t) where f (u) can be any twice-differentiable function.

5 Proof that f (x ± vt) Solves the Wave Equation Write f (x ± vt) as f (u), where u = x ± vt. So u x 1 and u t v Use the chain rule: f f u x u x f u f f u t u t v f u Computing the second derivatives adds another factor of u/ x or u/ t: Substituting into the wave equation: f x f u f t f u v 2 2 f x 1 v f t f 1 2 f - v u v u 0

6 What does a typical wave look like? It s rigid and moves. Some waves (like water waves) change their shape in time, but, for this to occur, additional terms must occur in the wave equation. x

7 Sums of solutions: The wave equation is linear, so the principle of Superposition holds. If f 1 (x,t) and f 2 (x,t) are solutions to the wave equation, then f 1 (x,t) + f 2 (x,t) is also a solution. Proof: The derivative of a sum is the sum of the derivatives. ( f f ) f f x x x and ( ) f f t t t ( f1 f2) 1 ( f1 f2) f1 1 f 1 f2 1 f x v t x v t x v t 0 This means that waves can pass through each other. In doing so, they can interfere constructively or destructively.

8 The 1D Wave Equation for Light Waves E x E - t where E is the light electric field, is the permittivity, and is the permeability. We ll use cosine-plus-sine-wave solutions (harmonic waves): E ( x, t) B cos[ k( x v t)] C sin[ k( x v t)] Now, write: k (x ± vt) = k x ± k vt = k x ± t where = kv E ( x, t) B cos( kx t) C sin( kx t) where: k v 1 The speed of light in vacuum, usually called c, is cm/s.

9 A Simpler Equation for a Harmonic Wave E (x,t) = A cos[(kx ± t) q] Use the trigonometric identity: cos(z y) = cos(z) cos(y) + sin(z) sin(y) where z = kx ± t and y = q to obtain: E (x,t) = A cos(kx ± t) cos(q) + A sin(kx ± t) sin(q) which is the same result as before: E ( x, t) B cos( kx t) C sin( kx t) as long as: A cos(q) = B and A sin(q) = C

10 Amplitude and Absolute Phase For simplicity, use only the forwardpropagating wave: E (x,t) = A cos[(k x t ) q ] Define: A = Amplitude q = Absolute phase (or initial phase) Absolute phase = 0 For the time, t = 0: Absolute phase = 2p/3 A x 2p/k

11 Wavelength, etc. Spatial quantities: Wavelength l For a given time, t 0 : x k-vector magnitude: k = 2p/l wave number: k = 1/l = k/2p Period t For a given position, x 0 : Temporal quantities: Temporal quantities: t angular frequency: = 2p/t cyclical frequency: n = 1/t = /2p

12 How fast is the wave traveling? The Phase Velocity The phase velocity is the wavelength / period: v = l /t Since n = 1/t : l The wave moves one wavelength, l, in one period, t. x v = l v In terms of the k-vector magnitude, k = 2p /l, and the angular frequency, = 2p /t, this is: v = / k

13 The Phase of a Wave: the Instantaneous Frequency The phase f is everything inside the cosine. E (x,t) = A cos(f), where f(x,t) = k x t q f = f(x,t) and is not a constant, like q! What if f(x,t) is a more complicated function of x and t? One period t later: f(x,t) - 2p = f(x,t+t) Subtracting from f - 2p and dividing by t : 2π f( x, t t ) -f( x, t) - t t f t But 2p/t is the frequency, (t). So: (t) = - f / t Similarly: k = f / x (t) is called the instantaneous frequency.

14 Phase Delays It s also helpful to define a delay, T, that a wave experiences after propagating a distance, d: T = d / v l d The wave moves a distance, d, in the time, T. x This time can also be expressed as a phase delay, f, which is the difference in phase before and after moving the distance, d: f (x+d,t) - f (x,t) = [k (x+d) - t - q] - [k x - t - q] f = k d = T using k = / v and d = vt

15 Complex Numbers Consider a point, P = (x,y) on a 2D Cartesian co-ordinates. Let the x-coordinate be the real part and the y-coordinate the imaginary part of a complex number. y (Imaginary) x = A cos(f) P y = A sin(f) f x (Real) So, instead of using an ordered pair, (x,y), we can write: P = x + i y = A cos(f) + i A sin(f) where i -1

16 Euler s Formula exp(if) = cos(f) + i sin(f) So the point, P = A cos(f) + i A sin(f), can be written: P = A exp(if) where: A = Amplitude f = Phase Properties of exp(if): Re{exp(if)} = cos(f) Re{-i exp(if)} = sin(f)

17 Waves Using Complex Numbers The electric field of a light wave can be written: E (x,t) = A cos(kx t q) where A is real. Since exp(if) = cos(f) + i sin(f), E (x,t) can also be written: E (x,t) = Re { A exp[i(kx t q)] }

18 Waves Using Complex Amplitudes Define E(x,t) to be the complex field without the Re:, exp - -q E x t A i kx t, Aexp ( iq ) exp x - E x t - i k t where we ve separated the constants from the rapidly changing stuff. The resulting complex amplitude is: So: E0 Aexp( -iq ), exp - E x t E0 i kx t As written, this field is complex! How do you know if E 0 is real or complex? So always assume it s complex unless otherwise stated.

19 Complex numbers greatly simplify waves! Adding waves of the same frequency, but different amplitude and/or absolute phase, yields a wave of the same frequency. This addition is hard with trig functions, but easy with complex exps: E ( x, t) E exp i( kx - t) E exp i( kx - t) E exp i( kx - t) tot ( E E E )exp i( kx - t) where all amplitudes and absolute phases are lumped into E 1, E 2, and E 3. E E E E E E E 1 x or t But this only works for adding waves. Don t try to multiply them.

20 The 3D Wave Equation for the Electric Field and Its Solution A light wave can propagate in any direction in space. So we must allow the space derivative to be 3D: Ñ 2 E - me 2 E t 2 = 0 or: E E E E x y z t which has the solution: E(x, y,z,t) = E 0 exp[i(k r -w t)] where and k º ( k x, k y, k ) z k r º k x x + k y y + k z z k k k k r º ( x, y,z) x y z k is the k-vector (the propagation magnitude and direction) of the wave.

21 E 0 exp[i(k r -wt)] is called a plane wave. A plane wave s contours of maximum field, called wave-fronts or phase-fronts, are planes. They extend over all space. Wave-fronts are helpful for drawing pictures of interfering waves. k A wave s wavefronts sweep along at the speed of light. A plane wave s wave-fronts are equally spaced, a wavelength apart. They re perpendicular to the propagation direction. Usually, we just draw lines; it s easier.

22 Longitudinal vs. Transverse Waves The direction of the wave s variations is called its polarization. Longitudinal: Motion is along the direction of propagation longitudinal polarization. Transverse: Motion is transverse to the direction of propagation transverse polarization. Light will always have transverse polarization. Space has 3 dimensions; 2 are transverse to the propagation direction, so there are 2 transverse waves and 1 potential longitudinal one.

23 Vector Fields A light wave has both electric and magnetic 3D vector fields: Electric field E B Magnetic field y x z

24 The 3D wave equation for the electric field is actually a vector equation. A light-wave electric field can point in any direction in space (as long as it s to k r ): Ñ 2 E - me 2 E t 2 = 0 Note the arrow over the E. which has the solution: E(x, y,z,t) = E 0 exp[i(k r -w t)] where and and k º ( k x, k y, k ) z k r º k x x + k y y + k z z E 0 = (E 0x, E 0 y, E 0z ) r º ( x, y,z)

25 Waves Using Complex Vector Amplitudes We must now allow the complex field E and its amplitude E 0 to be vectors: ( ) = E 0 expéi( k r -wt) E r,t ë ù û Note the arrows over the E s. In principle, the complex vector amplitude has six real numbers that must be specified to completely determine it: x-component y-component z-component E 0 = (Re{E 0x }+ iim{e 0x }, Re{E 0 y }+ iim{e 0 y }, Re{E 0z }+ iim{e 0z }) Not all six numbers can be nonzero for a given light wave, however, because the field will be perpendicular to the propagation direction.