2. Waves, the Wave Equation, and Phase Velocity · ... the Wave Equation, and Phase ... To displace...
Transcript of 2. Waves, the Wave Equation, and Phase Velocity · ... the Wave Equation, and Phase ... To displace...
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Waves, the Wave Equation, and Phase VelocityWhat is a wave?
Forward [f(x-vt)] and backward [f(x+vt)] propagating waves
The one-dimensional wave equation
Wavelength, frequency, period, etc.
Phase velocity Complex numbers
Plane waves and laser beams Boundary conditions
Div, grad, curl, etc., and the 3D Wave equation
f(x)f(x-3)
f(x-2)f(x-1)
x0 1 2 3
Source: Trebino, Georgia Tech
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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.
If we let a = v t, where v is positive and t is time, then the displacement will increase with time.
So represents a rightward, or forward, propagating wave.
Similarly, represents a leftward, or backward, propagating wave.
v will be the velocity of the wave.
f(x)f(x-3)
f(x-2)f(x-1)
x0 1 2 3
f(x - v t)
f(x + v t)
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The one-dimensional wave equation
2 2
2 2 2
1 0v
f fx t
∂ ∂− =
∂ ∂
The one-dimensional wave equation for scalar (i.e., non-vector) functions, f:
where v will be the velocity of the wave.
( , ) ( v )f x t f x t= ±
The wave equation has the simple solution:
where f (u) can be any twice-differentiable function.
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Proof that f (x ± vt) solves the wave equation
Write 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
∂=
∂vu
t∂
= ±∂
f f ux u x
∂ ∂ ∂=
∂ ∂ ∂f f ut u t
∂ ∂ ∂=
∂ ∂ ∂
f fx u
∂ ∂=
∂ ∂vf f
t u∂ ∂
= ±∂ ∂
2 22
2 2vf ft u
∂ ∂=
∂ ∂
2 2
2 2
f fx u
∂ ∂=
∂ ∂
2 2 2 22
2 2 2 2 2 2
1 1 v 0v v
f f f fx t u u
∂ ∂ ∂ ∂− = − = ∂ ∂ ∂ ∂
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The 1D wave equation for light waves
We’ll use cosine- and sine-wave solutions:
or
where:
2 2
2 2 0E Ex t
µε∂ ∂− =
∂ ∂
( , ) cos[ ( v )] sin[ ( v )]E x t B k x t C k x t= ± + ±
( , ) cos( ) sin( )E x t B kx t C kx tω ω= ± + ±
1vkω
µε= =
( v)kx k t±
where E is the light electric field
The speed of light in vacuum, usually called “c”, is 3 x 1010 cm/s.
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A simpler equation for a harmonic wave:
E(x,t) = A cos[(kx – ωt) – θ]
Use the trigonometric identity:
cos(z–y) = cos(z) cos(y) + sin(z) sin(y)
where z = kx – ω t and y = θ to obtain:
E(x,t) = A cos(kx – ωt) cos(θ) + A sin(kx – ωt) sin(θ)
which is the same result as before,
as long as:A cos(θ) = B and A sin(θ) = C
( , ) cos( ) sin( )E x t B kx t C kx tω ω= − + −For simplicity, we’ll just use the forward-propagating wave.
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Definitions: Amplitude and Absolute phase
E(x,t) = A cos[(k x – ω t ) – θ ]
A = Amplitudeθ = Absolute phase (or initial phase)
π
kx
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Definitions
Spatial quantities:
Temporal quantities:
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The Phase Velocity
How fast is the wave traveling?
Velocity is a reference distancedivided by a reference time.
The phase velocity is the wavelength / period: v = λ / τ
Since ν = 1/τ :
In terms of the k-vector, k = 2π / λ, and the angular frequency, ω = 2π / τ, this is:
v = λ v
v = ω / k
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The phase is everything inside the cosine.
E(x,t) = A cos(ϕ), where ϕ = k x – ω t – θ
ϕ = ϕ(x,y,z,t) and is not a constant, like θ !
In terms of the phase,
ω = – ∂ϕ /∂t
k = ∂ϕ /∂xAnd
– ∂ϕ /∂tv = –––––––
∂ϕ /∂x
The Phase of a Wave
This formula is useful when the wave is really complicated.
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Tacoma Narrows Bridge
1. The animation shows the Tacoma Narrows Bridge shortly before its collapse. What is its frequency?A .1 HzB .25 HzC .50 HzD 1 Hz
2. 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 torsional wave?A 1720 mB 860 mC 430 mD There is no way to tell.
3. What is the amplitude?A 0.4 mB 4 mC 8 mD 16 m
Animation: http://www.youtube.com/watch?v=3mclp9QmCGs
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Complex numbers
So, instead of using an ordered pair, (x,y), we write:
P = x + i y= A cos(ϕ) + i A sin(ϕ)
where i = (-1)1/2
Consider a point,P = (x,y), on a 2D Cartesian grid.
Let the x-coordinate be the real part and the y-coordinate the imaginary part of a complex number.
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Euler's Formula
exp(iϕ) = cos(ϕ) + i sin(ϕ)
so the point, P = A cos(ϕ) + i A sin(ϕ), can be written:
P = A exp(iϕ)
where
A = Amplitude
ϕ = Phase
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Proof of Euler's Formula
Use Taylor Series:
2 3 4
2 4 3
exp( ) 1 ...1! 2! 3! 4!
1 ... ...2! 4! 1! 3!
cos( ) sin( )
i ii
i
i
ϕ ϕ ϕ ϕϕ
ϕ ϕ ϕ ϕ
ϕ ϕ
= + − − + +
= − + + + − +
= +
2 3
( ) (0) '(0) ''(0) '''(0) ...1! 2! 3!x x xf x f f f f= + + + +
exp(iϕ) = cos(ϕ) + i sin(ϕ)
2 3 4
2 4 6 8
3 5 7 9
exp( ) 1 ...1! 2! 3! 4!
cos( ) 1 ...2! 4! 6! 8!
sin( ) ...1! 3! 5! 7! 9!
x x x xx
x x x xx
x x x x xx
= + + + + +
= − + − + +
= − + − + +
If we substitute x = iϕinto exp(x), then:
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Complex number theorems
[ ]
[ ]
[ ][ ]
1 1 2 2 1 2 1 2
1 1 2 2 1 2 1 2
exp( ) 1 exp( / 2) exp(- ) cos( ) sin( )
1 cos( ) exp( ) exp( )21 sin( ) exp( ) exp( )2
exp( ) exp( ) exp ( )
exp( ) / exp( ) / exp ( )
ii ii i
i i
i ii
A i A i A A i
A i A i A A i
ππϕ ϕ ϕ
ϕ ϕ ϕ
ϕ ϕ ϕ
ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ
= −=
= −
= + −
= − −
× = +
= −
exp( ) cos( ) sin( )i iϕ ϕ ϕ= +If
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More complex number theoremsAny complex number, z, can be written:
z = Re{ z } + i Im{ z }So
Re{ z } = 1/2 ( z + z* )and
Im{ z } = 1/2i ( z – z* )
where z* 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 }2
To convert z into polar form, A exp(iϕ):
A2 = Re{ z }2 + Im{ z }2
tan(ϕ) = Im{ z } / Re{ z }
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We can also differentiate exp(ikx) as if the argument were real.
[ ]
[ ]
exp( ) exp( )
cos( ) sin( ) sin( ) cos( )
1 sin( ) cos( )
1/ sin( ) cos( )
d ikx ik ikxdx
d kx i kx k kx ik kxdx
ik kx kxi
i i ik i kx kx
=
+ = − +
= − +
− = = +
Proof :
But , so :
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Waves using complex numbers
The electric field of a light wave can be written:
E(x,t) = A cos(kx – ωt – θ)
Since exp(iϕ) = cos(ϕ) + i sin(ϕ), E(x,t) can also be written:
E(x,t) = Re { A exp[i(kx – ωt – θ)] }
orE(x,t) = 1/2 A exp[i(kx – ωt – θ)] + c.c.
where "+ c.c." means "plus the complex conjugate of everything before the plus sign."
We often write these
expressions without the
½, Re, or +c.c.
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Waves using complex amplitudesWe 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
ex, ex( pp )
E x t A i kx t
E x t i k ti xA
ω θ
ωθ
= − −
= −−
0 exp( ) E A iθ= − ←
(note the " ~ ")
( ) ( )0, expE x t E i kx tω= −
How do you know if E0 is real or complex?
Sometimes people use the "~", but not always.So always assume it's complex.
As written, this entire field is complex!
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Complex numbers simplify waves!
This isn't so obvious using trigonometric functions, but it's easywith complex exponentials:
1 2 3
1 2 3
( , ) exp ( ) exp ( ) exp ( ) ( ) exp ( )
totE x t E i kx t E i kx t E i kx tE E E i kx t
ω ω ωω
= − + − + −= + + −
where all initial phases are lumped into E1, E2, and E3.
Adding waves of the same frequency, but different initial phase, yields a wave of the same frequency.
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is called a plane wave.
A plane wave's wave-fronts are equally spaced, a wavelength apart.
They're perpendicular to the propagation direction.
Wave-fronts are helpful for drawing pictures of interfering
waves.
A wave's wave-fronts sweep along at the
speed of light.
A plane wave’s contours of maximum field, called wave-fronts or phase-fronts, are planes. They extend over all space.
0 exp[ ( )]E i kx tω−
Usually, we just draw lines; it’s
easier.
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Localized waves in space: beamsA plane wave has flat wave-fronts throughout all space. It also has infinite energy.It doesn’t exist in reality.
Real waves are more localized. We can approximate a realistic wave as a plane wave vs. z times a Gaussian in x and y:
2 2
20( , , , ) exp[ ( )exp ]x yE x y z t E i kz tw
ω= +−
−
Laser beam spot on wall
w
x
y
Localized wave-fronts
z
x
exp(-x2)
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Localized waves in time: pulses
If we can localize the beam in space by multiplying by a Gaussian in x and y, we can also localize it in time by multiplying by a Gaussian in time.
22 2
220( , , , ) exp exp[ (exp )]x yE x y z t E i kzt twτ
ω +
−
= − −
t
E
This is the equation for a laser pulse.
t
exp(-t2)
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Longitudinal vs. Transverse waves
Motion is along the direction of propagation—longitudinal polarization
Motion is transverse to the direction of propagation—transverse polarization
Space has 3 dimensions, of which 2 are transverse to the propagation direction, so there are 2 transverse waves in addition to the potential longitudinal one.The direction of the wave’s variations is called its polarization.
Transverse:
Longitudinal:
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Vector fieldsLight is a 3D vector field.
A 3D vector field assigns a 3D vector (i.e., an arrow having both direction and length) to each point in 3D space.
( )f r
A light wave has both electric and magnetic 3D vector fields:
Wind patterns: 2D vector field
And it can propagate in any direction.
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Div, Grad, Curl, and all that
Types of 3D vector derivatives:
The Del operator:
The Gradient of a scalar function f :
The gradient points in the direction of steepest ascent.
, ,x y z
∂ ∂ ∂∇ ≡ ∂ ∂ ∂
, ,f f ffx y z
∂ ∂ ∂∇ ≡ ∂ ∂ ∂
If you want toknow more aboutvector calculus,read this book!
Div, Grad, Curl, and All That: An Informal Text on Vector Calculus , by Schey
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Div, Grad, Curl, and all that
The Divergence of a vector function:
yx zff ff
x y z∂∂ ∂
∇ ⋅ ≡ + +∂ ∂ ∂
The Divergence is nonzero if there are sources or sinks.
A 2D source with a large divergence:
Note that the x-component of this function changes rapidly in the x direction, etc., the essence of a large divergence.
xy
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Div, Grad, Curl, and more all thatThe Laplacian of a scalar function :
The Laplacian of a vector function is the same, but for each component of f:
2 , ,f f ff fx y z
∂ ∂ ∂∇ ≡ ∇ ⋅∇ = ∇ ⋅ ∂ ∂ ∂
2 2 2
2 2 2
f f fx y z
∂ ∂ ∂= + +
∂ ∂ ∂
2 2 22 2 2 2 2 22
2 2 2 2 2 2 2 2 2, ,y y yx x x z z zf f ff f f f f ffx y z x y z x y z
∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂∇ = + + + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
The Laplacian tells us the curvature of a vector function.
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The 3D wave equation for the electric field and its solution
or
whose solution is:
where
and
2 2 2 2
2 2 2 2 0E E E Ex y z t
µε∂ ∂ ∂ ∂+ + − =
∂ ∂ ∂ ∂
{ }0( , , , ) Re exp[ ( )]E x y z t E i k r tω= ⋅ −
x y zk r k x k y k z⋅ ≡ + +
2 2 2 2x y zk k k k≡ + +
22
2 0EEt
µε ∂∇ − =
∂
A light wave can propagate in any direction in space. So we must allow the space derivative to be 3D:
( ), ,x y zk k k k≡
( ), ,r x y z≡
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The 3D wave equation for a light-wave electric field is actually a vector equation.
whose solution is:
where:
{ }0( , , , ) Re exp[ ( )]E x y z t E i k r tω= ⋅ −
22
2 0EEt
µε ∂∇ − =
∂
And a light-wave electric field can point in any direction in space:
0 0 0 0( , , )x y zE E E E=
Note the arrow over the E.
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We must now allow the field E and its complex field amplitude to be vectors:
Vector Waves
( ) ( ){ }0, Re expE r t E i k r tω = ⋅ −
0 (Re{ } Im{ }, Re{ } Im{ }, Re{ } Im{ })x x y y z zE E i E E i E E i E= + + +
The complex vector amplitude has six numbers that must be specified to completely determine it!
0E
x-component y-component z-component
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Boundary Conditions
Often, a wave is constrained by external factors, which we call Boundary Conditions.
For example, a guitar string is attached at both ends.
In this case, only certain wavelengths/frequencies are possible.
Here the wavelengths can be:
λ1, λ1/2, λ1/3, λ1/4, etc.
Node Anti-node