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Mussel’s contraction biophysics. Kinematics and dynamics rotation
motion. Oscillation and wave. Mechanical wave. Acoustics.
Waves, the Wave Equation, and Phase Velocity
What is a wave?
Forward [f(x-vt)] vs. backward [f(x+vt)] propagating waves
The one-dimensional wave equation
Phase velocity
Complex numbers
What is a wave?A wave is anything that moves.
To displace any function f(x)to the right, just change itsargument 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 f(x-vt) represents a rightward, or forward,propagating wave.
Similarly, f(x+vt) represents a leftward, or backward,propagating wave.
v will be the velocity of the wave.
The one-dimensional wave equation
where f(u) can be any twice-differentiable function.
2 2
2 2 2
10
v
f f
x t
( , ) ( v )f x t f x t
has the simple solution:
Proof that f(x±vt) solves the wave equation
1. Write f(x±vt) as f(u), where u=x±vt. So and
2. Now, use the chain rule:
3. So and
4. Substituting into the wave equation:
1u
x
vu
t
f f u
x u x
f f u
t u t
f f
x u
vf f
t u
2 2
22 2
vf f
t u
2 2
2 2
f f
x u
2 2 2 22
2 2 2 2 2 2
1 1v 0
v v
f f f f
x t u u
QED
The 1D wave equation for light waves
We’ll use cosine- and sine-wave solutions:
or
where:
2 2
2 20
E E
x t
( , ) 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 t
1v
k
A simpler equation for a harmonic wave:
E(x,t) = A cos([kx – t] – )
Use the trigonometric identity:
cos(u–v) = cos(u)cos(v) + sin(u)sin(v)
to obtain:
E(x,t) = A cos(kx – t) cos() + A sin(kx – t) sin()
which is the same result as before, as long as:
A cos() = B and A sin() = C
Life will get even simpler in a few minutes!
Definitions: Amplitude and Absolute phase
E(x,t) = A cos([kx – t] – )
A = Amplitude
= Absolute phase (or initial phase)
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 = /
In terms of the k-vector, k = 2/ , and the angular frequency, = 2/ , this is:
v = / k
The Phase of a Wave
The phase is everything inside the cosine.
E(t) = A cos(), where = kx – t –
We give the phase a special name because it comes up so often.
In terms of the phase,
= – /t
k = /x
and
– /t
v = –––––––
/x
Electromagnetism is linear: The principle of “Superposition” holds.
If E1(x,t) and E2(x,t) are solutions to Maxwell’s equations,
then E1(x,t) + E2(x,t) is also a solution.
Proof: and
Typically, one sine wave plus another equals a sine wave.
This means that light beams can pass through each other.
It also means that waves can constructively or destructively interfere.
2 2 21 2 1 2
2 2 2
( )E E E E
x x x
2 2 21 2 1 2
2 2 2
( )E E E E
t t t
2 2 2 2 2 21 2 1 2 1 1 2 2
2 2 2 2 2 2
( ) ( )0
E E E E E E E E
x t x t x t
Complex numbers
So, instead of using an ordered pair, (x,y), we write:
P = x + i y = A cos() + i A sin()
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.
Euler's Formula
exp(i) = cos() + i sin()
so the point, P = A cos() + i A sin(), can be
written:
P = A exp(i)
where
A = Amplitude
= Phase
Proof of Euler's Formula exp(i) = cos() + i sin()
Using Taylor Series:2 3
2 3 4
2 4 6 8
3 5 7 9
( ) (0) '(0) ''(0) '''(0) ...1! 2! 3!
So:
exp( ) 1 ...1! 2! 3! 4!
cos( ) 1 ...2! 4! 6! 8!
sin( ) ...1! 3! 5! 7! 9!
If we subsitute x=i into exp(x), then:
exp( ) 11!
x x xf x f f f f
x x x xx
x x x xx
x x x x xx
ii
2 3 4
2 4 3
...2! 3! 4!
1 ... ...2! 4! 1! 3!
cos( ) sin( )
i
i
i
Complex number theorems
1 1 2 2 1 2 1 2
1 1 2
If exp( ) cos( ) sin( )
then:
exp( ) 1
exp( / 2)
exp(- ) cos( ) sin( )
1 cos( ) exp( ) exp( )
21
sin( ) exp( ) exp( )2
exp( ) exp( ) exp ( )
exp( ) / exp(
i i
i
i i
i i
i i
i ii
A i A i A A i
A i A
2 1 2 1 2) / exp ( )i A A i
More complex number theorems
1. Any 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 )
2. The "magnitude," |z|, of a complex number is:
|z|2 = z z*
3. To convert z into polar form, A exp(i):
A2 = Re{ z }2 + Im{ z }2
tan() = Im{ z } / Re{ z }
We can also differentiate exp(ikx) as if the argument were real.
exp( ) exp( )
Proof:
cos( ) sin( ) sin( ) cos( )
1 sin( ) cos( )
But 1/ , so: sin( ) cos( )
dikx ik ikx
dx
dkx i kx k kx ik kx
dx
ik kx kxi
i i ik i kx kx
QED
Waves using complex numbers
The electric field of a light wave can be written:
E(x,t) = A cos(kx – t – )
Since exp(i) = cos() + i sin(), E(x,t) can also be written:
E(x,t) = Re { A exp[i(kx – t – )] }
or
E(x,t) = 1/2 A exp[i(kx – t – )] } + c.c.
where "+ c.c." means "plus the complex conjugate of everything before the plus sign."
The 3D wave equation for the electric field
or
which has the solution:
where
and
2 2 2 2
2 2 2 20
E E E E
x y z t
( , , , ) exp[ ( )]E x y z t A i k r t
x y zk r k x k y k z
2 2 2 2x y zk k k k
22
20
EE
t
Waves using complex amplitudes
We can let the amplitude be complex:
where we've separated out the constant stuff from the rapidly changing stuff.
The resulting "complex amplitude" is:
So:
How do you know if E0 is real or complex?
Sometimes people use the "~", but not always.So always assume it's complex.
, exp
, [ exp( )] exp
E r t A i k r t
E r t A i i k r t
~ 0
~ ~ 0
exp( ) (note the "~")
, exp
E A i
E r t E i k r t