Download 2. Waves, the Wave Equation, and Phase Velocity

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 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 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
2f
 x2

1 2f
v2  t 2
 0
has the simple solution:
f ( x, t )  f ( x  vt )
where f(u) can be any twice-differentiable function.
Proof that f(x±vt) solves the wave equation
u
1. Write f(x±vt) as f(u), where u=x±vt. So  u  1 and
v
x
2. Now, use the chain rule:
3. So
 f  f u

 x u  x
t
 f  f u

t u t
2
2
f f
f
f
2f

f 2f
2  f


and

v
v

2
2
2
 x u
t
u
t
 u2
x
u
4. Substituting into the wave equation:
2f
1 2f
 2
2
x
v  t2
2f
1  22f 

 2 v
 0
2
2 
u
v  u 
QED
The 1D wave equation for light waves
 2E
 2E
  2  0
2
x
t
We’ll use cosine- and sine-wave solutions:
E ( x, t )  B cos(k[ x  vt ])  C sin(k[ x  vt ])
or
E ( x, t )  B cos(kx   t )  C sin(kx   t )
where:

k
 v 
1

A simpler equation for a harmonic wave:
E(x,t) = A cos([kx – t] – q)
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(q) + A sin(kx – t) sin(q)
which is the same result as before, as long as:
A cos(q) = B and
A sin(q) = C
Life will get even simpler in a few minutes!
Definitions: Amplitude and Absolute phase
E(x,t) = A cos([kx – t] – q)
A = Amplitude
q = Absolute phase (or initial phase)
Definitions
Spatial quantities:
Temporal quantities:
The Phase Velocity
How fast is the wave traveling? Velocity is a reference distance
divided by a reference time.
The phase velocity is the wavelength / period:
v = l/t
In terms of the k-vector, k = 2p/ l, and
the angular frequency,  = 2p/ t, this is:
v = /k
The Phase of a Wave
The phase is everything inside the cosine.
E(t) = A cos(j), where j = kx – t – q
We give the phase a special name because it comes up so often.
In terms of the phase,
 = – j/t
k = j/x
and
– j/t
v = –––––––
j/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:
 2 ( E1  E2 )  2 E1  2 E2
 2 ( E1  E2 )  2 E1  2 E2


and


2
2
2
2
2
t
t
 t2
x
x
x
 2 ( E1  E2 )
 2 ( E1  E2 )   2 E1
 2 E1    2 E2
 2 E2 
 
  2  

 
0
2
2
2 
2
2 
x
t
t    x
t 
x
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.
Complex numbers
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.
So, instead of using an ordered pair, (x,y), we write:
P = x+iy
= A cos(j) + i A sin(j)
where i = (-1)^1/2
Euler's Formula
exp(ij) = cos(j) + i sin(j)
so the point, P = A cos(j) + i A sin(j), can be
written:
P = A exp(ij)
where
A = Amplitude
j = Phase
Proof of Euler's Formula
exp(ij) = cos(j) + i sin(j)
Using Taylor Series:
x
x2
x3
f ( x)  f (0)  f '(0) 
f ''(0) 
f '''(0)  ...
1!
2!
3!
So:
x x 2 x3 x 4
exp( x)  1      ...
1! 2! 3! 4!
x 2 x 4 x 6 x8
cos( x)  1      ...
2! 4! 6! 8!
x x3 x5 x 7 x9
sin( x)       ...
1! 3! 5! 7! 9!
If we subsitute x=ij into exp(x), then:
ij j 2 ij 3 j 4
exp(ij )  1  


 ...
1! 2! 3! 4!
 j2 j4
 j j 3

 1 

 ...  i    ...
 2! 4!
  1! 3!

 cos(j )  i sin(j )
Complex number theorems
If exp(ij )  cos(j )  i sin(j )
then:
exp(ip )  1
exp(ip / 2)  i
exp(-ij )  cos(j )  i sin(j )
1
cos(j )   exp(ij )  exp( ij ) 
2
1
sin(j )   exp(ij )  exp( ij ) 
2i
A1exp(ij1 )  A2 exp(ij 2 )  A1 A2 exp i (j1  j 2 ) 
A1exp(ij1 ) / A2 exp(ij 2 )  A1 / A2 exp i (j1  j 2 ) 
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(ij):
A2 = Re{ z }2 + Im{ z }2
tan(j) = Im{ z } / Re{ z }
We can also differentiate exp(ikx) as if
the argument were real.
d
exp(ikx)  ik exp(ikx)
dx
Proof:
d
cos(kx)  i sin(kx)  k sin(kx)  ik cos(kx)
dx
 1

 ik   sin(kx)  cos(kx) 
 i

But  1/ i  i, so:
 ik i sin(kx)  cos(kx) 
QED
Waves using complex numbers
The electric field of a light wave can be written:
E(x,t) = A cos(kx – t – q)
Since exp(ij) = cos(j) + i sin(j), E(x,t) can also be written:
E(x,t) = Re { A exp[i(kx – t – q)] }
or
E(x,t) = 1/2 A exp[i(kx – t – q)] } + c.c.
where "+ c.c." means "plus the complex
conjugate of everything before the plus sign."
The 3D wave equation for the electric field
2

E
2
 E   2  0
t
or
 2E  2E  2E
 2E


  2  0
2
2
2
x
y
z
t
which has the solution:
E ( x, y, z, t )  A exp[i (k  r   t  q )]
where
and
k  r  kx x  k y y  kz z
k  k k k
2
2
x
2
y
2
z
Waves using complex amplitudes
We can let the amplitude be complex:
 


E  r , t   [ A exp(iq )] exp i  k  r   t  


E r , t  A exp i k  r   t  q
where we've separated out the constant stuff from the rapidly
changing stuff.
The resulting "complex amplitude" is:
E  A exp(iq )
~0
So:
 
(note the "~")

E r , t  E exp i k  r   t
~
~0

How do you know if E0 is real or complex?
Sometimes people use the "~", but not always.
So always assume it's complex.