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)

Definitions

Spatial quantities:

Temporal quantities:

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