5. Superposition of Waves 5. Superposition of Waves

5. Superposition of Waves 5. Superposition of Waves

Last Lecture

• Wave Equations

• Harmonic Waves

• Plane Waves

• EM Waves This Lecture

• Superposition of Waves

Superposition of Waves Superposition of Waves

1 2

2 2 2 2

2

2 2 2 2 2

1 2

1 2

:

1 v

. ,

,

Suppose that and are both solutions of the wave equation

x y z t

Then any linear combination of and is

also a solution of the wave equation For example if a and b are constants then

a b

is a

ψ ψ

ψ ψ ψ ψ ψ

ψ ψ

ψ ψ ψ

∂ ∂ ∂ ∂

+ + = ∇ =

∂ ∂ ∂ ∂

= +

. lso a solution of the wave equation

E. Hecht, Optics, Chapter 7.

5-2. Superposition of Waves of the Same Frequency 5-2. Superposition of Waves of the Same Frequency

1 2

01

cos(

)

cos(

)

E

E E

E α ω t E α ω t

= +

= − + −

Constructive interference and Destructive interference

Constructive interference and Destructive interference

Constructive interference :

( )

( )

2 1

1 2 01 1 02 1

01 02 1

2

cos( ) cos( 2 )

cos( )

m

E E E t E m t

E E t

α α π

α ω α π ω

α ω

− =

+ = − + + −

= + −

Destructive interference :

( )

( )

( )

2 1

1 2 01 1 02 1

01 02 1

2 1

cos( ) cos( 2 1 )

cos( )

m

E E E t E m t

E E t

α α π

α ω α π ω

α ω

− = +

+ = − + + + −

= − −

General superposition of the Same Frequency General superposition of the Same Frequency

( )

( )

1 2

1 2

( ) ( )

01 02

0 01 02

( )

0 0

Re

Defining

Re cos( )

i t i t

R

i i

i

i t

R

E E e E e

E e E e E e

E E e E t

α ω α ω

α α

α

α ω

α ω

− −

−

= +

≡ +

⇒ = = −

2 2

0 01 02 01 02 2 1

01 1 02 2

01 1 02 2

2 cos( )

sin sin

tan cos cos

E E E E E

E E

E E

α α

α α

α α α

= + + −

= +

+

Superposition of Multiple Waves of the Same Frequency Superposition of Multiple Waves of the Same Frequency

( ) ( ) ( )

( )

0 0

1

2 2

0 0 0 0

1 1

0 1

0 1

:

, cos cos

2 cos

sin tan

cos

N

i i

i

N N N

i i j j i

i j i i

N

i i

i N

i i

i

The relations just developed can be extended to the addition of an arbitrary number N of waves

E x t E t E t

E E E E

E E

α ω α ω

α α

α α

α

=

= > =

=

=

= − = −

= + −

=

∑

∑ ∑∑

∑

∑

5-3. Random and coherent sources 5-3. Random and coherent sources

( )

2 2

0 0 0 0

1 1

2 cos

N N N

i i j j i

i j i i

E E E E α α

= > =

= ∑ + ∑∑ −

Randomly phased sources :

( )

2 2 2

0 0 0 0 0

1 1 1

2 cos

N N N N

i i j j i i

i j i i i

E E E E α α E

= > = =

= ∑ + ∑∑ − = ∑

1 N

i i

I I

=

= ∑

The resultant irradiance (intensity) of the randomly phased sources is the sum of the individual irradiances.

Coherent and in-phase sources :

( )

2 2 2

0 0 0 0 0 0 0

1 1 1 1

2 cos 2

N N N N N N

i i j j i i i j

i j i i i j i i

E E E E α α E E E

= > = = > =

= ∑ + ∑∑ − = ∑ + ∑∑

2

0 0

1 N

i i

I E

=

⎛ ⎞

= ⎜ ⎟

⎝ ∑ ⎠

The resultant irradiance (intensity) of the coherent sources is N² times the individual irradiances.

0 01

I = × N I

2

0 01

I = N × I

5-4. Standing waves 5-4. Standing waves

1 2 0

sin( )

sin( )

R R

E = E + E = E ω t + kx + E ω t − kx − ϕ

to the left to the right

2

cos( ) sin( )

2 2

R R

E

= E kx + ϕ ω t − ϕ

{ ϕ

= π } → E

= ( 2 E

sin kx ) cos ω t

Mirror reflection

kx 2 x π m π

= λ =

x m _{⎛ ⎞} λ 2

= ⎜ ⎟ ⎝ ⎠

Nodes at

In a laser cavity In a laser cavity

2 d m _⎛ λ

_⎞

= ⎜ ⎟

⎝ ⎠

2

m m

2

m

d c c

m m d

λ ν

λ

⎛ ⎞ ⎛ ⎞

= ⎜ ⎝ ⎟ ⎠ = = ⎜ ⎝ ⎟ ⎠

5-5. The beat phenomenon 5-5. The beat phenomenon

1 2

p

2

k k k = +

1 2

g

2

k k

k −

=

( )( )

( ) x 2 A cos k x

cos k x

ψ =

Superposition of Waves with Different Frequency Superposition of Waves with Different Frequency

: beat wavelength

k_p=

k_g =

If the 2 waves have approximately equal wavelengths,

Phase velocity and Group velocity Phase velocity and Group velocity

When the waves have also a time dependence,

1 2

1 2

2 2

p

g

ω ω ω ω ω ω

= +

= −

1 2

1 2

2 2

p

g

k k k

k k k

= +

= −

1 1 1

2 2 2

( , ) cos( )

( , ) cos( )

x t A k x t

x t A k x t

ψ ω

ψ ω

= −

= −

higher frequency wave

lower frequency wave (envelope)

phase velocity : ^{1 2}

1 2

p p

p

v k k k k

ω ω ω+ ω

= = ≈

+

1 2

1 2

g g

g

v d

k k k dk

ω ω ω− ω

= = ≈

group velocity : −

( )

( )

2 1

1 2 /

p

g p p

p p p

p

d d dv

v kv v k

dk dk dk

d c c dn k dn

v k v k v

dk n n dk n dk

v dn k

n d ω

λ π λ

λ

⎛ ⎞

= = = + ⎜ ⎟

⎝ ⎠

− ⎡ ⎤

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞

= + ⎜ ⎟⎝ ⎠= + ⎜⎝ ⎟⎜⎠⎝ ⎟⎠= ⎢⎣ − ⎜⎝ ⎟⎠⎥⎦

⎡ ⎛ ⎞⎤

= ⎢⎣ + ⎜⎝ ⎟⎠⎥⎦ ← =

Phase velocity and Group velocity Phase velocity and Group velocity

phase velocity : ^{1 2}

1 2

p p

p

v k k k k

ω ω ω+ ω

= = ≈

+ ^{1 2}

1 2

g g

g

v d

k k k dk

ω ω ω− ω

= = ≈

group velocity : −