5. Superposition of Waves5. 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 22

2 2 2 2 2

1 2

1 2

:

1v

. ,,

Suppose that and are both solutionsof the wave equation

x y z t

Then any linear combination of and isalso a solution of the wave equation For exampleif a and b are constants then

a b

is a

ψ ψ

ψ ψ ψ ψψ

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 1 02 2cos( ) cos( )RE E E

E t E tα ω α ω= += − + −

Constructive interference and Destructive interference Constructive interference

and Destructive interference

Constructive interference :

( )

( )

2 1

1 2 01 1 02 1

01 02 1

2cos( ) cos( 2 )

cos( )

mE E E t E m t

E E t

α α πα 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 tR

i ii

i tR

E E e E e

E e E e E e

E E e E t

α ω α ω

α αα

α ω α ω

− −

−

= +

≡ +

⇒ = = −

2 20 01 02 01 02 2 1

01 1 02 2

01 1 02 2

2 cos( )sin sintancos cos

E E E E EE EE E

α αα ααα α

= + + −

+=

+

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

( ) ( ) ( )

( )

0 01

2 20 0 0 0

1 1

01

01

:

, cos cos

2 cos

sin tan

cos

N

i ii

N N N

i i j j ii j i i

N

i iiN

i ii

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

E x t E t E t

E E E E

E

E

α ω α ω

α α

αα

α

=

= > =

=

=

= − = −

= + −

=

∑

∑ ∑∑

∑

∑

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

( )2 20 0 0 0

1 12 cos

N N N

i i j j ii j i i

E E E E α α= > =

= + −∑ ∑∑

Randomly phased sources :

( )2 2 20 0 0 0 0

1 1 12 cos

N N N N

i i j j i ii j i i i

E E E E Eα α= > = =

= + − =∑ ∑∑ ∑ 0 01

N

ii

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 20 0 0 0 0 0 0

1 1 1 12 cos 2

N N N N N N

i i j j i i i ji j i i i j i i

E E E E E E Eα α= > = = > =

= + − = +∑ ∑∑ ∑ ∑∑2

0 01

N

ii

I E=

⎛ ⎞= ⎜ ⎟⎝ ⎠∑

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

0 01I N I= ×

20 01I N I= ×

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

1 2 0 0sin( ) sin( )R RE E E E t kx E t kxω ω ϕ= + = + + − −to the left to the right

02 cos( )sin( )2 2R R

RE E kx tϕ ϕω= + −

{ } ( )02 sin cosR RE E kx tϕ π ω= → =

Mirror reflection

2 xkx mπ πλ

= =

2x m λ⎛ ⎞= ⎜ ⎟

⎝ ⎠Nodes at

In a laser cavity In a laser cavity

2md m λ⎛ ⎞= ⎜ ⎟

⎝ ⎠

2 2m m

m

d c cmm d

λ νλ

⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

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

1 2

2pk kk +

=

1 2

2gk kk −

=

( )( )( ) 2 cos cosg px A k x k xψ =

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

: beat wavelength

kp =

kg =

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 ωω

ω ωω

+=

−=

1 2

1 2

2

2

p

g

k kk

k kk

+=

−=

1 1 1

2 2 2

( , ) cos( )( , ) cos( )x t A k x tx t A k x t

ψ ωψ ω

= −= −

higher frequency wave

lower frequency wave (envelope)

phase velocity : 1 2

1 2

pp

p

vk k k kω ω ω ω+

= = ≈+

1 2

1 2

gg

g

dvk k k dkω ω ω ω−

= = ≈−

group velocity :

( )

( )

2 1

1 2 /

pg p p

p p p

p

dvd dv kv v kdk dk dk

d c c dn k dnv k v k vdk n n dk n dk

dnv kn d

ω

λ π λλ

⎛ ⎞= = = + ⎜ ⎟

⎝ ⎠− ⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= + = + = −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞= + ← =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

Phase velocity and Group velocity Phase velocity and Group velocity

phase velocity : 1 2

1 2

pp

p

vk k k kω ω ω ω+

= = ≈+

1 2

1 2

gg

g

dvk k k dkω ω ω ω−

= = ≈−

group velocity :